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Clusteryanam

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Nagasuri Bala Venkateswarlu
Nagasuri Bala Venkateswarlu

This talk is developed to address a refresher course at Yanam for one full day. I have introduced the audience to clustering, both hierarchical and non-hierarchical. Clustering methods such as K-Means, K-Mediods, etc all introduced with live demonstrations.

Engineering
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A Brief Dealing of Clustering Prof. Dr. N.B. Venkateswarlu Aditya Institute of Technology and Management Tekkali Venkat_ritch@yahoo.com
Cluster Analysis • What is Cluster Analysis? • Types of Data in Cluster Analysis • Partitioning Methods • Hierarchical Methods • Density-Based Methods • Cluster Evaluation • Grid-Based Methods • Model-Based Clustering Methods • Outlier Analysis • Summary
The Problem of Clustering • Given a set of points, with a notion of distance between points, group the points into some number of clusters, so that members of a cluster are in some sense as close to each other as possible.
What is Cluster Analysis? • Finding groups of objects such that the objects in a group will be similar (or related) to one another and different from (or unrelated to) the objects in other groups Inter-cluster distances are maximized Intra-cluster distances are minimized
What is Cluster Analysis? • Cluster: a collection of data objects – Similar to one another within the same cluster – Dissimilar to the objects in other clusters • Cluster analysis – Grouping a set of data objects into clusters • Clustering is unsupervised classification: no predefined classes • Typical applications – As a stand-alone tool to get insight into data distribution – As a preprocessing step for other algorithms
General Applications of Clustering • Pattern Recognition • Spatial Data Analysis – create thematic maps in GIS by clustering feature spaces – detect spatial clusters and explain them in spatial data mining • Image Processing • Economic Science (especially market research) • WWW – Document classification – Cluster Weblog data to discover groups of similar access patterns
Examples of Clustering Applications • Marketing: Help marketers discover distinct groups in their customer bases, and then use this knowledge to develop targeted marketing programs • Land use: Identification of areas of similar land use in an earth observation database • Insurance: Identifying groups of motor insurance policy holders with a high average claim cost • City-planning: Identifying groups of houses according to their house type, value, and geographical location • Earth-quake studies: Observed earth quake epicenters should be clustered along continent faults
Example: SkyCat • A catalog of 2 billion “sky objects” represented objects by their radiation in 9 dimensions (frequency bands). • Problem: cluster into similar objects, e.g., galaxies, nearby stars, quasars, etc. • Sloan Sky Survey is a newer, better version.
Example: Clustering CD’s (Collaborative Filtering) • Intuitively: music divides into categories, and customers prefer a few categories. – But what are categories really? • Represent a CD by the customers who bought it. • Similar CD’s have similar sets of customers, and vice-versa.
The Space of CD’s • Think of a space with one dimension for each customer. – Values in a dimension may be 0 or 1 only. • A CD’s point in this space is (x1, x2,…, xk), where xi = 1 iff the i th customer bought the CD. – Compare with the “correlated items” matrix: rows = customers; cols. = CD’s.
Example: Clustering Documents • Represent a document by a vector (x1, x2,…, xk), where xi = 1 iff the i th word (in some order) appears in the document. – It actually doesn’t matter if k is infinite; i.e., we don’t limit the set of words. • Documents with similar sets of words may be about the same topic.
Example: Protein Sequences • Objects are sequences of {C,A,T,G}. • Distance between sequences is edit distance, the minimum number of inserts and deletes needed to turn one into the other. • Note there is a “distance,” but no convenient space in which points “live.”
Notion of a Cluster can be Ambiguous How many clusters? Four ClustersTwo Clusters Six Clusters
Problems With Clustering • Clustering in two dimensions looks easy. • Clustering small amounts of data looks easy. • And in most cases, looks are not deceiving.
The Curse of Dimensionality • Many applications involve not 2, but 10 or 10,000 dimensions. • High-dimensional spaces look different: almost all pairs of points are at about the same distance. – Assuming random points within a bounding box, e.g., values between 0 and 1 in each dimension.
What Is Good Clustering? • A good clustering method will produce high quality clusters with – high intra-class similarity – low inter-class similarity • The quality of a clustering result depends on both the similarity measure used by the method and its implementation. • The quality of a clustering method is also measured by its ability to discover some or all of the hidden patterns.
Types of Clusterings • A clustering is a set of clusters • Important distinction between hierarchical and partitional sets of clusters • Partitional Clustering – A division data objects into non-overlapping subsets (clusters) such that each data object is in exactly one subset • Hierarchical clustering – A set of nested clusters organized as a hierarchical tree
Partitional Clustering Original Points A Partitional Clustering
Hierarchical Clustering p4 p1 p3 p2 p4p1 p2 p3 p4p1 p2 p3 Traditional Hierarchical Clustering Non-traditional Hierarchical Clustering Non-traditional Dendrogram Traditional Dendrogram
Other Distinctions Between Sets of Clusters • Exclusive versus non-exclusive – In non-exclusive clusterings, points may belong to multiple clusters. – Can represent multiple classes or ‘border’ points • Fuzzy versus non-fuzzy – In fuzzy clustering, a point belongs to every cluster with some weight between 0 and 1 – Weights must sum to 1 – Probabilistic clustering has similar characteristics • Partial versus complete – In some cases, we only want to cluster some of the data • Heterogeneous versus homogeneous – Cluster of widely different sizes, shapes, and densities
Types of Clusters • Well-separated clusters • Center-based clusters • Contiguous clusters • Density-based clusters • Property or Conceptual • Described by an Objective Function
Types of Clusters: Well- Separated • Well-Separated Clusters: – A cluster is a set of points such that any point in a cluster is closer (or more similar) to every other point in the cluster than to any point not in the cluster. 3 well-separated clusters
Types of Clusters: Center- Based • Center-based – A cluster is a set of objects such that an object in a cluster is closer (more similar) to the “center” of a cluster, than to the center of any other cluster – The center of a cluster is often a centroid, the average of all the points in the cluster, or a medoid, the most “representative” point of a cluster 4 center-based clusters
Types of Clusters: Contiguity- Based • Contiguous Cluster (Nearest neighbor or Transitive) – A cluster is a set of points such that a point in a cluster is closer (or more similar) to one or more other points in the cluster than to any point not in the cluster. 8 contiguous clusters
Types of Clusters: Density- Based • Density-based – A cluster is a dense region of points, which is separated by low- density regions, from other regions of high density. – Used when the clusters are irregular or intertwined, and when noise and outliers are present. 6 density-based clusters
Types of Clusters: Conceptual Clusters • Shared Property or Conceptual Clusters – Finds clusters that share some common property or represent a particular concept. 2 Overlapping Circles
Types of Clusters: Objective Function • Clusters Defined by an Objective Function – Finds clusters that minimize or maximize an objective function. – Enumerate all possible ways of dividing the points into clusters and evaluate the `goodness' of each potential set of clusters by using the given objective function. (NP Hard) – Can have global or local objectives. • Hierarchical clustering algorithms typically have local objectives • Partitional algorithms typically have global objectives – A variation of the global objective function approach is to fit the data to a parameterized model. • Parameters for the model are determined from the data. • Mixture models assume that the data is a ‘mixture' of a number of statistical distributions.
Types of Clusters: Objective Function … • Map the clustering problem to a different domain and solve a related problem in that domain – Proximity matrix defines a weighted graph, where the nodes are the points being clustered, and the weighted edges represent the proximities between points – Clustering is equivalent to breaking the graph into connected components, one for each cluster. – Want to minimize the edge weight between clusters and maximize the edge weight within clusters
Requirements of Clustering in Data Mining • Scalability • Ability to deal with different types of attributes • Discovery of clusters with arbitrary shape • Minimal requirements for domain knowledge to determine input parameters • Able to deal with noise and outliers • Insensitive to order of input records • High dimensionality • Incorporation of user-specified constraints • Interpretability and usability
Data Structures • Data matrix – (two modes) • Dissimilarity matrix – (one mode)                   npx...nfx...n1x ............... ipx...ifx...i1x ............... 1px...1fx...11x                 0...)2,()1,( ::: )2,3() ...ndnd 0dd(3,1 0d(2,1) 0
Measure the Quality of Clustering • Dissimilarity/Similarity metric: Similarity is expressed in terms of a distance function, which is typically metric: d(i, j) • There is a separate “quality” function that measures the “goodness” of a cluster. • The definitions of distance functions are usually very different for interval-scaled, boolean, categorical, ordinal and ratio variables. • Weights should be associated with different variables based on applications and data semantics. • It is hard to define “similar enough” or “good enough” – the answer is typically highly subjective.
Type of data in clustering analysis • Interval-scaled variables: • Binary variables: • Nominal, ordinal, and ratio variables: • Variables of mixed types:
Interval-valued variables • Standardize data – Calculate the mean absolute deviation: where – Calculate the standardized measurement (z-score) • Using mean absolute deviation is more robust than using standard deviation .)... 21 1 nffff xx(xnm +++= |)|...|||(|1 21 fnffffff mxmxmxns −++−+−= f fif if s mx z − =
Similarity and Dissimilarity Between Objects • Distances are normally used to measure the similarity or dissimilarity between two data objects • Some popular ones include: Minkowski distance: where i = (xi1, xi2, …, xip) and j = (xj1, xj2, …, xjp) are two p-dimensional data objects, and q is a positive integer • If q = 1, d is Manhattan distance q q pp qq j x i x j x i x j x i xjid )||...|||(|),( 2211 −++−+−= ||...||||),( 2211 pp j x i x j x i x j x i xjid −++−+−=
Similarity and Dissimilarity Between Objects (Cont.) • If q = 2, d is Euclidean distance: – Properties • d(i,j) ≥ 0 • d(i,i) = 0 • d(i,j) = d(j,i) • d(i,j) ≤ d(i,k) + d(k,j) • Also, one can use weighted distance, parametric Pearson product moment correlation, or other disimilarity measures )||...|||(|),( 22 22 2 11 pp j x i x j x i x j x i xjid −++−+−=
Distance Measures • Each clustering problem is based on some kind of “distance” between points. • Two major classes of distance measure: 1. Euclidean 2. Non-Euclidean
Euclidean Vs. Non- Euclidean • A Euclidean space has some number of real-valued dimensions and “dense” points. – There is a notion of “average” of two points. – A Euclidean distance is based on the locations of points in such a space. • A Non-Euclidean distance is based on properties of points, but not their “location” in a space.
Axioms of a Distance Measure • d is a distance measure if it is a function from pairs of points to reals such that: 1. d(x,y) > 0. 2. d(x,y) = 0 iff x = y. 3. d(x,y) = d(y,x). 4. d(x,y) < d(x,z) + d(z,y) (triangle inequality ).
Some Euclidean Distances • L2 norm : d(x,y) = square root of the sum of the squares of the differences between x and y in each dimension. – The most common notion of “distance.” • L1 norm : sum of the differences in each dimension. – Manhattan distance = distance if you had to travel along coordinates only.
Examples of Euclidean Distances x = (5,5) y = (9,8) L2-norm: dist(x,y) = √(42 +32 ) = 5 L1-norm: dist(x,y) = 4+3 = 7 4 35
Another Euclidean Distance • L∞ norm : d(x,y) = the maximum of the differences between x and y in any dimension. • Note: the maximum is the limit as n goes to ∞ of what you get by taking the n th power of the differences, summing and taking the n th root.
Non-Euclidean Distances • Jaccard distance for sets = 1 minus ratio of sizes of intersection and union. • Cosine distance = angle between vectors from the origin to the points in question. • Edit distance = number of inserts and deletes to change one string into another.
Jaccard Distance • Example: p1 = 10111; p2 = 10011. – Size of intersection = 3; size of union = 4, Jaccard measure (not distance) = 3/4. • Need to make a distance function satisfying triangle inequality and other laws. • d(x,y) = 1 – (Jaccard measure) works.
Why J.D. Is a Distance Measure • d(x,x) = 0 because x∩x = x∪x. • d(x,y) = d(y,x) because union and intersection are symmetric. • d(x,y) > 0 because |x∩y| < |x∪y|. • d(x,y) < d(x,z) + d(z,y) trickier --- next slide.
Triangle Inequality for J.D. 1 - |x ∩z| + 1 - |y ∩z| > 1 - |x ∩y| |x ∪z| |y ∪z| |x ∪y| • Remember: |a ∩b|/|a ∪b| = probability that minhash(a) = minhash(b). • Thus, 1 - |a ∩b|/|a ∪b| = probability that minhash(a) ≠ minhash(b).
Triangle Inequality --- (2) • So we need to observe that prob[minhash(x) ≠ minhash(y)] < prob[minhash(x) ≠ minhash(z)] + prob[minhash(z) ≠ minhash(y)] • Clincher: whenever minhash(x) ≠ minhash(y), one of minhash(x) ≠ minhash(z) and minhash(z) ≠ minhash(y) must be true.
Cosine Distance • Think of a point as a vector from the origin (0,0,…,0) to its location. • Two points’ vectors make an angle, whose cosine is the normalized dot-product of the vectors: p1.p2/|p2||p1|. – Example p1 = 00111; p2 = 10011. – p1.p2 = 2; |p1| = |p2| = √3. – cos(θ) = 2/3; θ is about 48 degrees.
Cosine-Measure Diagram p1 p2p1.p2 θ |p2| dist(p1, p2) = θ = arccos(p1.p2/|p2||p1|)
Why? p1 = (x1,y1) p2 = (x2,0)x1 θ Dot product is invariant under rotation, so pick convenient coordinate system. x1 = p1.p2/|p2| p1.p2 = x1*x2. |p2| = x2.
Why C.D. Is a Distance Measure • d(x,x) = 0 because arccos(1) = 0. • d(x,y) = d(y,x) by symmetry. • d(x,y) > 0 because angles are chosen to be in the range 0 to 180 degrees. • Triangle inequality: physical reasoning. If I rotate an angle from x to z and then from z to y, I can’t rotate less than from x to y.
Edit Distance • The edit distance of two strings is the number of inserts and deletes of characters needed to turn one into the other. • Equivalently, d(x,y) = |x| + | y| -2|LCS(x,y)|. – LCS = longest common subsequence = longest string obtained both by deleting from x and deleting from y.
Example • x = abcde ; y = bcduve. • Turn x into y by deleting a, then inserting u and v after d. – Edit-distance = 3. • Or, LCS(x,y) = bcde. • |x| + |y| - 2|LCS(x,y)| = 5 + 6 –2*4 = 3.
Why E.D. Is a Distance Measure • d(x,x) = 0 because 0 edits suffice. • d(x,y) = d(y,x) because insert/delete are inverses of each other. • d(x,y) > 0: no notion of negative edits. • Triangle inequality: changing x to z and then to y is one way to change x to y.
Variant Edit Distance • Allow insert, delete, and mutate. – Change one character into another. • Minimum number of inserts, deletes, and mutates also forms a distance measure.
Binary Variables • A contingency table for binary data • Simple matching coefficient (invariant, if the binary variable is symmetric): • Jaccard coefficient (noninvariant if the binary variable is asymmetric): dcba cbjid +++ +=),( pdbcasum dcdc baba sum ++ + + 0 1 01 cba cbjid ++ +=),( Object i Object j
Dissimilarity between Binary Variables • Example – gender is a symmetric attribute – the remaining attributes are asymmetric binary – let the values Y and P be set to 1, and the value N be set to 0 Name Gender Fever Cough Test-1 Test-2 Test-3 Test-4 Jack M Y N P N N N Mary F Y N P N P N Jim M Y P N N N N 75.0 211 21 ),( 67.0 111 11 ),( 33.0 102 10 ),( = ++ + = = ++ + = = ++ + = maryjimd jimjackd maryjackd
Nominal Variables • A generalization of the binary variable in that it can take more than 2 states, e.g., red, yellow, blue, green • Method 1: Simple matching – m: # of matches, p: total # of variables • Method 2: use a large number of binary variables – creating a new binary variable for each of the M nominal states p mpjid −=),(
Ordinal Variables • An ordinal variable can be discrete or continuous • Order is important, e.g., rank • Can be treated like interval-scaled – replace xif by their rank – map the range of each variable onto [0, 1] by replacing i-th object in the f-th variable by – compute the dissimilarity using methods for interval-scaled variables 1 1 − − = f if if M r z },...,1{ fif Mr ∈
Ratio-Scaled Variables • Ratio-scaled variable: a positive measurement on a nonlinear scale, approximately at exponential scale, such as AeBt or Ae-Bt • Methods: – treat them like interval-scaled variables—not a good choice! (why?—the scale can be distorted) – apply logarithmic transformation yif = log(xif) – treat them as continuous ordinal data treat their rank as interval- scaled
Variables of Mixed Types • A database may contain all the six types of variables – symmetric binary, asymmetric binary, nominal, ordinal, interval and ratio • One may use a weighted formula to combine their effects – f is binary or nominal: dij (f) = 0 if xif = xjf , or dij (f) = 1 o.w. – f is interval-based: use the normalized distance – f is ordinal or ratio-scaled • compute ranks rif and • and treat zif as interval-scaled )( 1 )()( 1 ),( f ij p f f ij f ij p f d jid δ δ = = Σ Σ = 1 1 − − = f if M rzif
Cluster Analysis • What is Cluster Analysis? • Types of Data in Cluster Analysis • Partitioning Methods • Hierarchical Methods • Density-Based Methods • Cluster Evaluation • Grid-Based Methods • Model-Based Clustering Methods • Outlier Analysis • Summary
Partitioning Algorithms: Basic Concept • Partitioning method: Construct a partition of a database D of n objects into a set of k clusters • Given a k, find a partition of k clusters that optimizes the chosen partitioning criterion – Global optimal: exhaustively enumerate all partitions – Heuristic methods: k-means and k-medoids algorithms – k-means (MacQueen’67): Each cluster is represented by the center of the cluster – k-medoids or PAM (Partition around medoids) (Kaufman & Rousseeuw’87): Each cluster is represented by one of the objects in the cluster
The K-Means Clustering Method • Given k, the k-means algorithm is implemented in four steps: – Partition objects into k nonempty subsets – Compute seed points as the centroids of the clusters of the current partition (the centroid is the center, i.e., mean point, of the cluster) – Assign each object to the cluster with the nearest seed point – Go back to Step 2, stop when no more new assignment
The K-Means Clustering Method 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 K=2 Arbitrarily choose K object as initial cluster center Assign each objects to most similar center Update the cluster means Update the cluster means reassignreassign
k –Means Algorithm(s) • Assumes Euclidean space. • Start by picking k, the number of clusters. • Initialize clusters by picking one point per cluster. – For instance, pick one point at random, then k -1 other points, each as far away as possible from the previous points.
Populating Clusters 1. For each point, place it in the cluster whose current centroid it is nearest. 2. After all points are assigned, fix the centroids of the k clusters. 3. Reassign all points to their closest centroid. – Sometimes moves points between clusters.
Example 1 2 3 4 5 6 7 8x x Clusters after first round Reassigned points
Getting k Right – Try different k, looking at the change in the average distance to centroid, as k increases. • Average falls rapidly until right k, then changes little. k Average distance to centroid Best value of k
Example x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x x x Too few; many long distances to centroid.
Example x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x x x Just right; distances rather short.
Example x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x x x Too many; little improvement in average distance.
BFR Algorithm • BFR (Bradley-Fayyad-Reina) is a variant of k -means designed to handle very large (disk-resident) data sets. • It assumes that clusters are normally distributed around a centroid in a Euclidean space. – Standard deviations in different dimensions may vary.
BFR --- (2) • Points are read one main-memory-full at a time. • Most points from previous memory loads are summarized by simple statistics. • To begin, from the initial load we select the initial k centroids by some sensible approach.
Initialization: k -Means • Possibilities include: 1. Take a small sample and cluster optimally. 2. Take a sample; pick a random point, and then k – 1 more points, each as far from the previously selected points as possible.
Three Classes of Points 1. The discard set : points close enough to a centroid to be represented statistically. 2. The compression set : groups of points that are close together but not close to any centroid. They are represented statistically, but not assigned to a cluster. 3. The retained set : isolated points.
Representing Sets of Points • For each cluster, the discard set is represented by: 1. The number of points, N. 2. The vector SUM, whose i th component is the sum of the coordinates of the points in the i th dimension. 3. The vector SUMSQ: i th component = sum of squares of coordinates in i th dimension.
Comments • 2d + 1 values represent any number of points. – d = number of dimensions. • Averages in each dimension (centroid coordinates) can be calculated easily as SUMi /N. – SUMi = i th component of SUM.
Comments --- (2) • Variance of a cluster’s discard set in dimension i can be computed by: (SUMSQi /N ) – (SUMi /N )2 • And the standard deviation is the square root of that. • The same statistics can represent any compression set.
“Galaxies” Picture A cluster. Its points are in the DS. The centroid Compressed sets. Their points are in the CS. Points in the RS
Processing a “Memory- Load” of Points 1. Find those points that are “sufficiently close” to a cluster centroid; add those points to that cluster and the DS. 2. Use any main-memory clustering algorithm to cluster the remaining points and the old RS. Clusters go to the CS; outlying points to the RS.
Processing --- (2) 3. Adjust statistics of the clusters to account for the new points. 4. Consider merging compressed sets in the CS. 5. If this is the last round, merge all compressed sets in the CS and all RS points into their nearest cluster.
A Few Details . . . • How do we decide if a point is “close enough” to a cluster that we will add the point to that cluster? • How do we decide whether two compressed sets deserve to be combined into one?
How Close is Close Enough? • We need a way to decide whether to put a new point into a cluster. • BFR suggest two ways: 1. The Mahalanobis distance is less than a threshold. 2. Low likelihood of the currently nearest centroid changing.
Mahalanobis Distance • Normalized Euclidean distance. • For point (x1,…,xk) and centroid (c1,…,ck): 1. Normalize in each dimension: yi = |xi -ci|/σi 2. Take sum of the squares of the yi ’s. 3. Take the square root.
Mahalanobis Distance --- (2) • If clusters are normally distributed in d dimensions, then one standard deviation corresponds to a distance √d. – I.e., 70% of the points of the cluster will have a Mahalanobis distance < √d. • Accept a point for a cluster if its M.D. is < some threshold, e.g. 4 standard deviations.
Picture: Equal M.D. Regions σ 2σ
Should Two CS Subclusters Be Combined? • Compute the variance of the combined subcluster. – N, SUM, and SUMSQ allow us to make that calculation. • Combine if the variance is below some threshold.
Comments on the K-Means Method • Strength: Relatively efficient: O(tkn), where n is # objects, k is # clusters, and t is # iterations. Normally, k, t << n. • Comparing: PAM: O(k(n-k)2 ), CLARA: O(ks2 + k(n-k)) • Comment: Often terminates at a local optimum. The global optimum may be found using techniques such as: deterministic annealing and genetic algorithms • Weakness – Applicable only when mean is defined, then what about categorical data? – Need to specify k, the number of clusters, in advance – Unable to handle noisy data and outliers – Not suitable to discover clusters with non-convex shapes
Variations of the K-Means Method • A few variants of the k-means which differ in – Selection of the initial k means – Dissimilarity calculations – Strategies to calculate cluster means • Handling categorical data: k-modes (Huang’98) – Replacing means of clusters with modes – Using new dissimilarity measures to deal with categorical objects – Using a frequency-based method to update modes of clusters – A mixture of categorical and numerical data: k-prototype method
What is the problem of k- Means Method? • The k-means algorithm is sensitive to outliers ! – Since an object with an extremely large value may substantially distort the distribution of the data. • K-Medoids: Instead of taking the mean value of the object in a cluster as a reference point, medoids can be used, which is the most centrally located object in a cluster. 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10
Importance of Choosing Initial Centroids -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 0.5 1 1.5 2 2.5 3 x y Iteration 1 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 0.5 1 1.5 2 2.5 3 x y Iteration 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 0.5 1 1.5 2 2.5 3 x y Iteration 3 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 0.5 1 1.5 2 2.5 3 x y Iteration 4 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 0.5 1 1.5 2 2.5 3 x y Iteration 5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 0.5 1 1.5 2 2.5 3 x y Iteration 6
Solutions to Initial Centroids Problem • Multiple runs – Helps, but probability is not on your side • Sample and use hierarchical clustering to determine initial centroids • Select more than k initial centroids and then select among these initial centroids – Select most widely separated • Postprocessing • Bisecting K-means – Not as susceptible to initialization issues
Limitations of K-means: Differing Density Original Points K-means (3 Clusters)
Limitations of K-means: Non- globular Shapes Original Points K-means (2 Clusters)
The CURE Algorithm • Problem with BFR/k -means: – Assumes clusters are normally distributed in each dimension. – And axes are fixed --- ellipses at an angle are not OK. • CURE: – Assumes a Euclidean distance. – Allows clusters to assume any shape.
Example: Stanford Faculty Salaries e e e e e e e e e e e h h h h h h h h h h h h h salary age
Starting CURE 1. Pick a random sample of points that fit in main memory. 2. Cluster these points hierarchically --- group nearest points/clusters. 3. For each cluster, pick a sample of points, as dispersed as possible. 4. From the sample, pick representatives by moving them (say) 20% toward the centroid of the cluster.
Example: Initial Clusters e e e e e e e e e e e h h h h h h h h h h h h h salary age
Example: Pick Dispersed Points e e e e e e e e e e e h h h h h h h h h h h h h salary age Pick (say) 4 remote points for each cluster.
Example: Pick Dispersed Points e e e e e e e e e e e h h h h h h h h h h h h h salary age Move points (say) 20% toward the centroid.
Finishing CURE • Now, visit each point p in the data set. • Place it in the “closest cluster.” – Normal definition of “closest”: that cluster with the closest (to p ) among all the sample points of all the clusters.
Curse of Dimensionality • One way to look at it: in large-dimension spaces, random vectors are perpendicular. Why? Argument #1: Lots of 2-dim subspaces. There must be one where the vectors’ projections are almost perpendicular. Argument #2: Expected value of cosine of angle is 0.
Cosine of Angle Between Random Vectors • Assume vectors emanate from the origin (0,0,…,0). • Components are random in range [-1,1]. • (a1,a2,…,an).(b1,b2,…,bn) has expected value 0 and a standard deviation that grows as √n. • But lengths of both vectors grow as √n. • So dot product around √n/ (√n * √n) = 1/√n.
Random Vectors --- Continued • Thus, a typical pair of vectors has an angle whose cosine is on the order of 1/√n. • As n -> ∞, that’s 0; i.e., the angle is about 90°.
Interesting Consequence • Suppose “random vectors are perpendicular,” even in non-Euclidean spaces. • Suppose we know the distance from A to B, say d (A,B ), and we also know d (B,C ), but we don’t know d (A,C ). • Suppose B and C are fairly close, say in the same cluster. • What is d (A,C )?
Diagram of Situation A C B Approximately perpendicular Assuming points lie in a plane: d (A,B )2 + d (B,C )2 = d (A,C )2
Important Point • Why do we assume AB is perpendicular to AC, and not that either of the other two angles are right- angles? 1. AB and AC are not “random vectors”; they each go to points that are far away from A and close to each other. 2. If AB is longer than AC, then it is angle ACB that is right, but both ACB and ABC are approximately right-angles.
Dealing With a Non- Euclidean Space • Problem: clusters cannot be represented by centroids. • Why? Because the “average” of “points” might not be a point in the space. • Best substitute: the clustroid = point in the cluster that minimizes the sum of the squares of distances to the points in the cluster.
Representing Clusters in Non- Euclidean Spaces • Recall BFR represents a Euclidean cluster by N, SUM, and SUMSQ. • A non-Euclidean cluster is represented by: – N. – The clustroid. – Sum of the squares of the distances from clustroid to all points in the cluster.
Example of CoD Use • Problem: in non-Euclidean space, we want to decide whether to merge two clusters. – Each cluster represented by N, clustroid, and “SUMSQ.” – Also, SUMSQ for each point in the cluster, even if it is not the clustroid. • Merge if SUMSQ for new cluster is “low.”
Estimating SUMSQ p p ’s clustroid, c other clust- roid, b
Suppose p Were the Clustroid of Combined Cluster • It’s SUMSQ would be the sum of: 1. Old SUMSQ(p) [for old cluster containing p]. 2. SUMSQ(b) plus d (p,b)2 times number of points in b ’s cluster. • Critical point: vector p ->b assumed perpendicular to vectors from b to all other points in its cluster --- justifies (2).
Combining Clusters --- Continued • We can thus estimate SUMSQ for each point in the combined cluster. Take the point with the least SUMSQ as the clustroid of the new cluster --- provided that SUMSQ is small enough.
The GRGPF Algorithm • From Ganti et al. --- see reading list. • Works for non-Euclidean distances. • Works for massive (disk-resident) data. • Hierarchical clustering. • Clusters are grouped into a tree of disk blocks (like a B-tree or R-tree).
Information Retained About a Cluster 1. N, clustroid, SUMSQ. 2. The p points closest to the clustroid, and their values of SUMSQ. 3. The p points of the cluster that are furthest away from the clustroid, and their SUMSQ’s.
At Interior Nodes of the Tree • Interior nodes have samples of the clustroids of the clusters found at descendant leaves of this node. • Try to keep clusters on one leaf block close, descendants of a level-1 node close, etc. • Interior part of tree kept in main memory.
Picture of the Tree cluster data cluster data samples main memory on disk
Initialization • Take a main-memory sample of points. • Organize them into clusters hierarchically. • Build the initial tree, with level-1 interior nodes representing clusters of clusters, and so on. • All other points are inserted into this tree.
Inserting Points • Start at the root. • At each interior node, visit one or more children that have sample clustroids near the inserted point. • At the leaves, insert the point into the cluster with the nearest clustroid.
Updating Cluster Data • Suppose we add point X to a cluster. • Increase count N by 1. • For each of the 2p + 1 points Y whose SUMSQ is stored, add d (X,Y )2 . • Estimate SUMSQ for X.
Estimating SUMSQ(X ) • If C is the clustroid, SUMSQ(X ) is, by the CoD assumption: Nd (X,C ) 2 + SUMSQ(C ) – Based on assumption that vector from X to C is perpendicular to vectors from C to all the other nodes of the cluster. • This value may allow X to replace one of the closest or furthest nodes.
Possible Modification to Cluster Data • There may be a new clustroid --- one of the p closest points --- because of the addition of X. • Eventually, the clustroid may migrate out of the p closest points, and the entire representation of the cluster needs to be recomputed.
Splitting and Merging Clusters • Maintain a threshold for the radius of a cluster = √(SUMSQ/N ). • Split a cluster whose radius is too large. • Adding clusters may overflow leaf blocks, and require splits of blocks up the tree. – Splitting is similar to a B-tree. – But try to keep locality of clusters.
Splitting and Merging --- (2) • The problem case is when we have split so much that the tree no longer fits in main memory. • Raise the threshold on radius and merge clusters that are sufficiently close.
Merging Clusters • Suppose there are nearby clusters with clustroids C and D, and we want to consider merging them. • Assume that the clustroid of the combined cluster will be one of the p furthest points from the clustroid of one of those clusters.
Merging --- (2) • Compute SUMSQ(X ) [from the cluster of C ] for the combined cluster by summing: 1. SUMSQ(X ) from its own cluster. 2. SUMSQ(D ) + N [d (X,C )2 + d (C,D )2 ]. Uses the CoD to reason that the distance from X to each point in the other cluster goes to C, makes a right angle to D, and another right angle to the point.
Merging --- Concluded • Pick as the clustroid for the combined cluster that point with the least SUMSQ. • But if this SUMSQ is too large, do not merge clusters. • Hope you get enough mergers to fit the tree in main memory.
Fastmap • Not a clustering algorithm --- rather, a method for applying multidimensional scaling. – That is, mapping the points onto a small- dimension space, so the CoD does not apply.
Fastmap --- (2) • Assumes non-Euclidean space. – But like GRGFP pretends it is working in 2- dimensional Euclidean space when it is convenient to do so. • Goal: map n points in much less than O(n 2 ) time. – I.e., you cannot compute distances between each pair of points and place points in k-dim. space to minimize error.
Fastmap --- Key Idea • Create a “dimension” in non-Euclidean space by: 1. Pick a pair of points A and B that are far apart. Start with random A; pick most distant B. 2. Treat AB as an “axis” and project all points onto AB, using the law of cosines.
Projecting Point C Onto AB A B C x d(A,C) d(B,C) d(A,B) x = [d 2 (A,C) + d 2 (A,B) – d 2 (B,C)]/2d (A,B)
Revising Distances • Having computed the position of every point along the pseudo-axis AB, we need to lower the distances between points in the “other dimensions.”
Picture A B C D dold(C,D) x y dnew(C,D) = √dold(C,D)2 – (x-y)2
But … • We can’t afford to compute new distances for each pseudo-dimension. – It would take O(n 2 ) time. • Rather, for each pseudo-dimension, store the position along the pseudo-axis for each point, and adjust the distance between points by square-subtract-sqrt only when needed. – I.e., one of the points is an axis-end.
Fastmap --- Summary • Pick a number of dimensions k. FOR i = 1 TO k DO BEGIN Pick a pseudo-axis AiBi; Compute projection of each point onto this pseudo-axis; END; • Each step is O(ni ); total O(nk 2 ).
The CURE Algorithm • Problem with BFR/k -means: – Assumes clusters are normally distributed in each dimension. – And axes are fixed --- ellipses at an angle are not OK. • CURE: – Assumes a Euclidean distance. – Allows clusters to assume any shape.
Example: Stanford Faculty Salaries e e e e e e e e e e e h h h h h h h h h h h h h salary age
Starting CURE 1. Pick a random sample of points that fit in main memory. 2. Cluster these points hierarchically --- group nearest points/clusters. 3. For each cluster, pick a sample of points, as dispersed as possible. 4. From the sample, pick representatives by moving them (say) 20% toward the centroid of the cluster.
Example: Initial Clusters e e e e e e e e e e e h h h h h h h h h h h h h salary age
Example: Pick Dispersed Points e e e e e e e e e e e h h h h h h h h h h h h h salary age Pick (say) 4 remote points for each cluster.
Example: Pick Dispersed Points e e e e e e e e e e e h h h h h h h h h h h h h salary age Move points (say) 20% toward the centroid.
Finishing CURE • Now, visit each point p in the data set. • Place it in the “closest cluster.” – Normal definition of “closest”: that cluster with the closest (to p ) among all the sample points of all the clusters.
Clustering a Stream (New Topic) • Assume points enter in a stream. • Maintain a sliding window of points. • Queries ask for clusters of points within some suffix of the window. • Only important issue: where are the cluster centroids. – There is no notion of “all the points” in a stream.
BDMO Approach • BDMO = Babcock, Datar, Motwani, O’Callaghan. • k –means based. • Can use less than O(N ) space for windows of size N. • Generalizes trick of DGIM: buckets of increasing “weight.”
Recall DGIM • Maintains a sequence of buckets B1, B2, … • Buckets have timestamps (most recent stream element in bucket). • Sizes of buckets nondecreasing. – In DGIM size = power of 2. • Either 1 or 2 of each size.
Alternative Combining Rule • Instead of “combine the 2nd and 3rd of any one size” we could say: • “Combine Bi+1 and Bi if size(Bi+1 ∪ Bi) < size(Bi-1 ∪ Bi-2 ∪ … ∪ B1).” – If Bi+1, Bi, and Bi-1 are the same size, inequality must hold (almost). – If Bi-1 is smaller, it cannot hold.
Buckets for Clustering • In place of “size” (number of 1’s) we use (an approximation to) the sum of the distances from all points to the centroid of their cluster. • Merge consecutive buckets if the “size” of the merged bucket is less than the sum of the sizes of all later buckets.
Consequence of Merge Rule • In a stable list of buckets, any two consecutive buckets are “bigger” than all smaller buckets. • Thus, “sizes” grow exponentially. • If there is a limit on total “size,” then the number of buckets is O(log N ). • N = window size. – E.g., all points are in a fixed hypercube.
Outline of Algorithm 1. What do buckets look like? – Clusters at various levels, represented by centroids. 2. How do we merge buckets? – Keep # of clusters at each level small. 3. What happens when we query? – Final clustering of all clusters of all relevant buckets.
Organization of Buckets • Each bucket consists of clusters at some number of levels. – 4 levels in our examples. • Clusters represented by: 1. Location of centroid. 2. Weight = number of points in the cluster. 3. Cost = upper bound on sum of distances from member points to centroid.
Processing Buckets --- (1) • Actions determined by N (window size) and k (desired number of clusters). • Also uses a tuning parameter τ for which we use 1/4 to simplify. – 1/τ is the number of levels of clusters.
Processing Buckets --- (2) • Initialize a new bucket with k new points. – Each is a cluster at level 0. • If the timestamp of the oldest bucket is outside the window, delete that bucket.
Level-0 Clusters • A single point p is represented by (p, 1, 0). • That is: 1. A point is its own centroid. 2. The cluster has one point. 3. The sum of distances to the centroid is 0.
Merging Buckets --- (1) • Needed in two situations: 1. We have to process a query, which requires us to (temporarily) merge some tail of the bucket sequence. 2. We have just added a new (most recent) bucket and we need to check the rule about two consecutive buckets being “bigger” than all that follow.
Merging Buckets --- (2) • Step 1: Take the union of the clusters at each level. • Step 2: If the number of clusters (points) at level 0 is now more than N 1/4 , cluster them into k clusters. – These become clusters at level 1. • Steps 3,…: Repeat, going up the levels, if needed.
Representing New Clusters • Centroid = weighted average of centroids of component clusters. • Weight = sum of weights. • Cost = sum over all component clusters of: 1. Cost of component cluster. 2. Weight of component times distance from its centroid to new centroid.
Example: New Centroid + (18,-2) + (3,3) + (12,12) centroids 5 10 15 weights + (12,2) new centroid
Example: New Costs + (18,-2) + (3,3) + (12,12) 5 10 15 + (12,2) old cost added true cost
Queries • Find all the buckets within the range of the query. – The last bucket may be only partially within the range. • Cluster all clusters at all levels into k clusters. • Return the k centroids.
Error in Estimation • Goal is to pick the k centroids that minimize the true cost (sum of distances from each point to its centroid). • Since recorded “costs” are inexact, there can be a factor of 2 error at each level. • Additional error because some of last bucket may not belong. – But fraction of spurious points is small (why?).
Effect of Cost-Errors 1. Alter when buckets get combined. Not really important. 2. Produce suboptimal clustering at any stage of the algorithm. The real measure of how bad the output is.
Speedup of Algorithm • As given, algorithm is slow. – Each new bucket causes O(log N ) bucket- merger problems. • A faster version allows the first bucket to have not k, but N 1/2 (or in general N 2τ ) points. – A number of consequences, including slower queries, more space.
The K-Medoids Clustering Method • Find representative objects, called medoids, in clusters • PAM (Partitioning Around Medoids, 1987) – starts from an initial set of medoids and iteratively replaces one of the medoids by one of the non-medoids if it improves the total distance of the resulting clustering – PAM works effectively for small data sets, but does not scale well for large data sets • CLARA (Kaufmann & Rousseeuw, 1990) • CLARANS (Ng & Han, 1994): Randomized sampling • Focusing + spatial data structure (Ester et al., 1995)
Typical k-medoids algorithm (PAM) 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Total Cost = 20 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 K=2 Arbitrary choose k object as initial medoids 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Assign each remainin g object to nearest medoids Randomly select a nonmedoid object,Oramdom Compute total cost of swapping 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Total Cost = 26 Swapping O and Oramdom If quality is improved. Do loop Until no change 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10
PAM (Partitioning Around Medoids) (1987) • PAM (Kaufman and Rousseeuw, 1987), built in Splus • Use real object to represent the cluster – Select k representative objects arbitrarily – For each pair of non-selected object h and selected object i, calculate the total swapping cost TCih – For each pair of i and h, • If TCih < 0, i is replaced by h • Then assign each non-selected object to the most similar representative object – repeat steps 2-3 until there is no change
PAM Clustering: Total swapping cost TCih=∑jCjih 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 j i h t Cjih = 0 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 t i h j Cjih = d(j, h) - d(j, i) 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 h i t j Cjih = d(j, t) - d(j, i) 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 t i h j Cjih = d(j, h) - d(j, t)
What is the problem with PAM? • Pam is more robust than k-means in the presence of noise and outliers because a medoid is less influenced by outliers or other extreme values than a mean • Pam works efficiently for small data sets but does not scale well for large data sets. – O(k(n-k)2 ) for each iteration where n is # of data,k is # of clusters Sampling based method, CLARA(Clustering LARge Applications)
CLARA (Clustering Large Applications) (1990) • CLARA (Kaufmann and Rousseeuw in 1990) – Built in statistical analysis packages, such as S+ • It draws multiple samples of the data set, applies PAM on each sample, and gives the best clustering as the output • Strength: deals with larger data sets than PAM • Weakness: – Efficiency depends on the sample size – A good clustering based on samples will not necessarily represent a good clustering of the whole data set if the sample is biased
CLARANS (“Randomized” CLARA) (1994) • CLARANS (A Clustering Algorithm based on Randomized Search) (Ng and Han’94) • CLARANS draws sample of neighbors dynamically • The clustering process can be presented as searching a graph where every node is a potential solution, that is, a set of k medoids • If the local optimum is found, CLARANS starts with new randomly selected node in search for a new local optimum • It is more efficient and scalable than both PAM and CLARA • Focusing techniques and spatial access structures may further improve its performance (Ester et al.’95)
Cluster Analysis • What is Cluster Analysis? • Types of Data in Cluster Analysis • Partitioning Methods • Hierarchical Methods • Density-Based Methods • Cluster Evaluation • Grid-Based Methods • Model-Based Clustering Methods • Outlier Analysis • Summary
Hierarchical Clustering • Use distance matrix as clustering criteria. This method does not require the number of clusters k as an input, but needs a termination condition Step 0 Step 1 Step 2 Step 3 Step 4 b d c e a a b d e c d e a b c d e Step 4 Step 3 Step 2 Step 1 Step 0 agglomerative (AGNES) divisive (DIANA)
AGNES (Agglomerative Nesting) • Introduced in Kaufmann and Rousseeuw (1990) • Implemented in statistical analysis packages, e.g., Splus • Use the Single-Link method and the dissimilarity matrix. • Merge nodes that have the least dissimilarity • Go on in a non-descending fashion • Eventually all nodes belong to the same cluster 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10
Agglomerative Clustering Algorithm • More popular hierarchical clustering technique • Basic algorithm is straightforward 1. Compute the proximity matrix 2. Let each data point be a cluster 3. Repeat 4. Merge the two closest clusters 5. Update the proximity matrix 6. Until only a single cluster remains • Key operation is the computation of the proximity of two clusters – Different approaches to defining the distance between clusters distinguish the different algorithms
Starting Situation • Start with clusters of individual points and a proximity matrix p1 p3 p5 p4 p2 p1 p2 p3 p4 p5 . . . . . . Proximity Matrix
Intermediate Situation • After some merging steps, we have some clusters C1 C4 C2 C5 C3 C2C1 C1 C3 C5 C4 C2 C3 C4 C5 Proximity Matrix
Intermediate Situation • We want to merge the two closest clusters (C2 and C5) and update the proximity matrix. C1 C4 C2 C5 C3 C2C1 C1 C3 C5 C4 C2 C3 C4 C5 Proximity Matrix
After Merging • The question is “How do we update the proximity matrix?” C1 C4 C2 U C5 C3 ? ? ? ? ? ? ? C2 U C5C1 C1 C3 C4 C2 U C5 C3 C4 Proximity Matrix
How to Define Inter-Cluster Similarity p1 p3 p5 p4 p2 p1 p2 p3 p4 p5 . . . . . . Similarity? • MIN • MAX • Group Average • Distance Between Centroids • Other methods driven by an objective function – Ward’s Method uses squared error Proximity Matrix
How to Define Inter-Cluster Similarity p1 p3 p5 p4 p2 p1 p2 p3 p4 p5 . . . . . . Proximity Matrix • MIN • MAX • Group Average • Distance Between Centroids • Other methods driven by an objective function – Ward’s Method uses squared error
How to Define Inter-Cluster Similarity p1 p3 p5 p4 p2 p1 p2 p3 p4 p5 . . . . . . Proximity Matrix • MIN • MAX • Group Average • Distance Between Centroids • Other methods driven by an objective function – Ward’s Method uses squared error
How to Define Inter-Cluster Similarity p1 p3 p5 p4 p2 p1 p2 p3 p4 p5 . . . . . . Proximity Matrix • MIN • MAX • Group Average • Distance Between Centroids • Other methods driven by an objective function – Ward’s Method uses squared error
How to Define Inter-Cluster Similarity p1 p3 p5 p4 p2 p1 p2 p3 p4 p5 . . . . . . Proximity Matrix • MIN • MAX • Group Average • Distance Between Centroids • Other methods driven by an objective function – Ward’s Method uses squared error × ×
Hierarchical Clustering: Time and Space requirements • O(N2 ) space since it uses the proximity matrix. – N is the number of points. • O(N3 ) time in many cases – There are N steps and at each step the size, N2 , proximity matrix must be updated and searched – Complexity can be reduced to O(N2 log(N) ) time for some approaches
Hierarchical Clustering: Problems and Limitations • Once a decision is made to combine two clusters, it cannot be undone • No objective function is directly minimized • Different schemes have problems with one or more of the following: – Sensitivity to noise and outliers – Difficulty handling different sized clusters and convex shapes – Breaking large clusters
A Dendrogram Shows How the Clusters are Merged Hierarchically • Decompose data objects into a several levels of nested partitioning (tree of clusters), called a dendrogram. • A clustering of the data objects is obtained by cutting the dendrogram at the desired level, then each connected component forms a cluster.
DIANA (Divisive Analysis) • Introduced in Kaufmann and Rousseeuw (1990) • Implemented in statistical analysis packages, e.g., Splus • Inverse order of AGNES • Eventually each node forms a cluster on its own 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10
More on Hierarchical Clustering Methods • Major weakness of agglomerative clustering methods – do not scale well: time complexity of at least O(n2 ), where n is the number of total objects – can never undo what was done previously • Integration of hierarchical with distance-based clustering – BIRCH (1996): uses CF-tree and incrementally adjusts the quality of sub-clusters – CURE (1998): selects well-scattered points from the cluster and then shrinks them towards the center of the cluster by a specified fraction – CHAMELEON (1999): hierarchical clustering using dynamic modeling
BIRCH (1996) • Birch: Balanced Iterative Reducing and Clustering using Hierarchies, by Zhang, Ramakrishnan, Livny (SIGMOD’96) • Incrementally construct a CF (Clustering Feature) tree, a hierarchical data structure for multiphase clustering – Phase 1: scan DB to build an initial in-memory CF tree (a multi-level compression of the data that tries to preserve the inherent clustering structure of the data) – Phase 2: use an arbitrary clustering algorithm to cluster the leaf nodes of the CF-tree • Scales linearly: finds a good clustering with a single scan and improves the quality with a few additional scans • Weakness: handles only numeric data, and sensitive to the order of the data record.
Clustering Feature: CF = (N, LS, SS) N: Number of data points LS: ∑N i=1=Xi SS: ∑N i=1=Xi 2 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 CF = (5, (16,30),(54,190)) (3,4) (2,6) (4,5) (4,7) (3,8) Clustering Feature Vector
CF-Tree in BIRCH • Clustering feature: – summary of the statistics for a given subcluster: the 0-th, 1st and 2nd moments of the subcluster from the statistical point of view. – registers crucial measurements for computing cluster and utilizes storage efficiently A CF tree is a height-balanced tree that stores the clustering features for a hierarchical clustering – A nonleaf node in a tree has descendants or “children” – The nonleaf nodes store sums of the CFs of their children • A CF tree has two parameters – Branching factor: specify the maximum number of children. – threshold: max diameter of sub-clusters stored at the leaf nodes
CF Tree CF1 child1 CF3 child3 CF2 child2 CF6 child6 CF1 child1 CF3 child3 CF2 child2 CF5 child5 CF1 CF2 CF6 prev next CF1 CF2 CF4 prev next B = 7 L = 6 Root Non-leaf node Leaf node Leaf node
CURE (Clustering Using REpresentatives ) • CURE: proposed by Guha, Rastogi & Shim, 1998 – Stops the creation of a cluster hierarchy if a level consists of k clusters – Uses multiple representative points to evaluate the distance between clusters, adjusts well to arbitrary shaped clusters and avoids single-link effect
Drawbacks of Distance- Based Method • Drawbacks of square-error based clustering method – Consider only one point as representative of a cluster – Good only for convex shaped, similar size and density, and if k can be reasonably estimated
Cure: The Algorithm • Draw random sample s. • Partition sample to p partitions with size s/p • Partially cluster partitions into s/pq clusters • Eliminate outliers – By random sampling – If a cluster grows too slow, eliminate it. • Cluster partial clusters. • Label data in disk
Data Partitioning and Clustering – s = 50 – p = 2 – s/p = 25 x x x y y y y x y x s/pq = 5
Cure: Shrinking Representative Points • Shrink the multiple representative points towards the gravity center by a fraction of α. • Multiple representatives capture the shape of the cluster x y x y
Clustering Categorical Data: ROCK • ROCK: Robust Clustering using linKs, by S. Guha, R. Rastogi, K. Shim (ICDE’99). – Use links to measure similarity/proximity – Not distance based – Computational complexity: • Basic ideas: – Similarity function and neighbors: Let T1 = {1,2,3}, T2={3,4,5} O n nm m n nm a( log )2 2 + + Sim T T T T T T ( , )1 2 1 2 1 2 = ∩ ∪ Sim T T( , ) { } { , , , , } .1 2 3 1 2 3 4 5 1 5 0 2= = =
CHAMELEON (Hierarchical clustering using dynamic modeling) • CHAMELEON: by G. Karypis, E.H. Han, and V. Kumar’99 • Measures the similarity based on a dynamic model – Two clusters are merged only if the interconnectivity and closeness (proximity) between two clusters are high relative to the internal interconnectivity of the clusters and closeness of items within the clusters – Cure ignores information about interconnectivity of the objects, Rock ignores information about the closeness of two clusters • A two-phase algorithm 1. Use a graph partitioning algorithm: cluster objects into a large number of relatively small sub-clusters 2. Use an agglomerative hierarchical clustering algorithm: find the genuine clusters by repeatedly combining these sub-clusters
Overall Framework of CHAMELEON Construct Sparse Graph Partition the Graph Merge Partition Final Clusters Data Set
Cluster Analysis • What is Cluster Analysis? • Types of Data in Cluster Analysis • Partitioning Methods • Hierarchical Methods • Density-Based Methods • Cluster Evaluation • Grid-Based Methods • Model-Based Clustering Methods • Outlier Analysis • Summary
Density-Based Clustering Methods • Clustering based on density (local cluster criterion), such as density-connected points • Major features: – Discover clusters of arbitrary shape – Handle noise – One scan – Need density parameters as termination condition • Several interesting studies: – DBSCAN: Ester, et al. (KDD’96) – OPTICS: Ankerst, et al (SIGMOD’99). – DENCLUE: Hinneburg & D. Keim (KDD’98) – CLIQUE: Agrawal, et al. (SIGMOD’98)
Density Concepts • Core object (CO)–object with at least ‘M’ objects within a radius ‘E-neighborhood’ • Directly density reachable (DDR)–x is CO, y is in x’s ‘E- neighborhood’ • Density reachable–there exists a chain of DDR objects from x to y • Density based cluster–density connected objects maximum w.r.t. reachability
Density-Based Clustering: Background • Two parameters: – Eps: Maximum radius of the neighbourhood – MinPts: Minimum number of points in an Eps-neighbourhood of that point • NEps(p): {q belongs to D | dist(p,q) <= Eps} • Directly density-reachable: A point p is directly density- reachable from a point q wrt. Eps, MinPts if – 1) p belongs to NEps(q) – 2) core point condition: |NEps (q)| >= MinPts p q MinPts = 5 Eps = 1 cm
Density-Based Clustering: Background (II) • Density-reachable: – A point p is density-reachable from a point q wrt. Eps, MinPts if there is a chain of points p1, …, pn, p1 = q, pn = p such that pi+1 is directly density- reachable from pi • Density-connected – A point p is density-connected to a point q wrt. Eps, MinPts if there is a point o such that both, p and q are density-reachable from o wrt. Eps and MinPts. p q p1 p q o
DBSCAN: Density Based Spatial Clustering of Applications with Noise • Relies on a density-based notion of cluster: A cluster is defined as a maximal set of density-connected points • Discovers clusters of arbitrary shape in spatial databases with noise Core Border Outlier Eps = 1cm MinPts = 5
DBSCAN • DBSCAN is a density-based algorithm. – Density = number of points within a specified radius (Eps) – A point is a core point if it has more than a specified number of points (MinPts) within Eps • These are points that are at the interior of a cluster – A border point has fewer than MinPts within Eps, but is in the neighborhood of a core point – A noise point is any point that is not a core point or a border point.
DBSCAN: Core, Border, and Noise Points
DBSCAN Algorithm • Eliminate noise points • Perform clustering on the remaining points
DBSCAN: Core, Border and Noise Points Original Points Point types: core, border and noise Eps = 10, MinPts = 4
When DBSCAN Works Well Original Points Clusters • Resistant to Noise • Can handle clusters of different shapes and sizes
When DBSCAN Does NOT Work Well Original Points (MinPts=4, Eps=9.75). (MinPts=4, Eps=9.92) • Varying densities • High-dimensional data
DBSCAN: Determining EPS and MinPts • Idea is that for points in a cluster, their kth nearest neighbors are at roughly the same distance • Noise points have the kth nearest neighbor at farther distance • So, plot sorted distance of every point to its kth nearest neighbor
OPTICS: A Cluster-Ordering Method (1999) • OPTICS: Ordering Points To Identify the Clustering Structure – Ankerst, Breunig, Kriegel, and Sander (SIGMOD’99) – Produces a special order of the database wrt its density-based clustering structure – This cluster-ordering contains info equiv to the density-based clusterings corresponding to a broad range of parameter settings – Good for both automatic and interactive cluster analysis, including finding intrinsic clustering structure – Can be represented graphically or using visualization techniques
Cluster Analysis • What is Cluster Analysis? • Types of Data in Cluster Analysis • Partitioning Methods • Hierarchical Methods • Density-Based Methods • Cluster Evaluation • Grid-Based Methods • Model-Based Clustering Methods • Outlier Analysis • Summary
Cluster Validity • For supervised classification we have a variety of measures to evaluate how good our model is – Accuracy, precision, recall • For cluster analysis, the analogous question is how to evaluate the “goodness” of the resulting clusters? • But “clusters are in the eye of the beholder”! • Then why do we want to evaluate them? – To avoid finding patterns in noise – To compare clustering algorithms – To compare two sets of clusters – To compare two clusters
Clusters found in Random Data 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x y Random Points 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x y K-means 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x y DBSCAN 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x y Complete Link
Different Aspects of Cluster Validation 1. Determining the clustering tendency of a set of data, i.e., distinguishing whether non-random structure actually exists in the data. 2. Comparing the results of a cluster analysis to externally known results, e.g., to externally given class labels. 3. Evaluating how well the results of a cluster analysis fit the data without reference to external information. - Use only the data 4. Comparing the results of two different sets of cluster analyses to determine which is better. 5. Determining the ‘correct’ number of clusters. For 2, 3, and 4, we can further distinguish whether we want to evaluate the entire clustering or just individual clusters.
Measures of Cluster Validity • Numerical measures that are applied to judge various aspects of cluster validity, are classified into the following three types. – External Index: Used to measure the extent to which cluster labels match externally supplied class labels. • Entropy – Internal Index: Used to measure the goodness of a clustering structure without respect to external information. • Sum of Squared Error (SSE) – Relative Index: Used to compare two different clusterings or clusters. • Often an external or internal index is used for this function, e.g., SSE or entropy • Sometimes these are referred to as criteria instead of indices – However, sometimes criterion is the general strategy and index is the numerical measure that implements the criterion.
Measuring Cluster Validity Via Correlation • Two matrices – Proximity Matrix – “Incidence” Matrix • One row and one column for each data point • An entry is 1 if the associated pair of points belong to the same cluster • An entry is 0 if the associated pair of points belongs to different clusters • Compute the correlation between the two matrices – Since the matrices are symmetric, only the correlation between n(n-1) / 2 entries needs to be calculated. • High correlation indicates that points that belong to the same cluster are close to each other. • Not a good measure for some density or contiguity based clusters.
Measuring Cluster Validity Via Correlation • Correlation of incidence and proximity matrices for the K-means clusterings of the following two data sets. 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x y 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x y Corr = -0.9235 Corr = -0.5810
Using Similarity Matrix for Cluster Validation • Order the similarity matrix with respect to cluster labels and inspect visually. 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x y Points Points 20 40 60 80 100 10 20 30 40 50 60 70 80 90 100 Similarity 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Using Similarity Matrix for Cluster Validation • Clusters in random data are not so crisp Points Points 20 40 60 80 100 10 20 30 40 50 60 70 80 90 100 Similarity 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 DBSCAN 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x y
Points Points 20 40 60 80 100 10 20 30 40 50 60 70 80 90 100 Similarity 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Using Similarity Matrix for Cluster Validation • Clusters in random data are not so crisp K-means 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 xy
Using Similarity Matrix for Cluster Validation 1 2 3 5 6 4 7 DBSCAN 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 500 1000 1500 2000 2500 3000 500 1000 1500 2000 2500 3000
Internal Measures: SSE • Clusters in more complicated figures aren’t well separated • Internal Index: Used to measure the goodness of a clustering structure without respect to external information – SSE • SSE is good for comparing two clusterings or two clusters (average SSE). • Can also be used to estimate the number of clusters 2 5 10 15 20 25 30 0 1 2 3 4 5 6 7 8 9 10 K SSE 5 10 15 -6 -4 -2 0 2 4 6
Internal Measures: SSE • SSE curve for a more complicated data set 1 2 3 5 6 4 7 SSE of clusters found using K-means
Framework for Cluster Validity • Need a framework to interpret any measure. – For example, if our measure of evaluation has the value, 10, is that good, fair, or poor? • Statistics provide a framework for cluster validity – The more “atypical” a clustering result is, the more likely it represents valid structure in the data – Can compare the values of an index that result from random data or clusterings to those of a clustering result. • If the value of the index is unlikely, then the cluster results are valid – These approaches are more complicated and harder to understand. • For comparing the results of two different sets of cluster analyses, a framework is less necessary. – However, there is the question of whether the difference between two index values is significant
Statistical Framework for SSE • Example – Compare SSE of 0.005 against three clusters in random data – Histogram shows SSE of three clusters in 500 sets of random data points of size 100 distributed over the range 0.2 – 0.8 for x and y values 0.016 0.018 0.02 0.022 0.024 0.026 0.028 0.03 0.032 0.034 0 5 10 15 20 25 30 35 40 45 50 SSE Count 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x y
Statistical Framework for Correlation • Correlation of incidence and proximity matrices for the K-means clusterings of the following two data sets. 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x y 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x y Corr = -0.9235 Corr = -0.5810
Internal Measures: Cohesion and Separation • Cluster Cohesion: Measures how closely related are objects in a cluster – Example: SSE • Cluster Separation: Measure how distinct or well-separated a cluster is from other clusters • Example: Squared Error – Cohesion is measured by the within cluster sum of squares (SSE) – Separation is measured by the between cluster sum of squares – Where |Ci| is the size of cluster i ∑ ∑ ∈ −= i Cx i i mxWSS 2 )( ∑ −= i ii mmCBSS 2 )(
Internal Measures: Cohesion and Separation • Example: SSE – BSS + WSS = constant 1 2 3 4 5 × ×× m1 m2 m 1091 9)35.4(2)5.13(2 1)5.45()5.44()5.12()5.11( 22 2222 =+= =−×+−×= =−+−+−+−= Total BSS WSSK=2 clusters: 10010 0)33(4 10)35()34()32()31( 2 2222 =+= =−×= =−+−+−+−= Total BSS WSSK=1 cluster:
Internal Measures: Cohesion and Separation • A proximity graph based approach can also be used for cohesion and separation. – Cluster cohesion is the sum of the weight of all links within a cluster. – Cluster separation is the sum of the weights between nodes in the cluster and nodes outside the cluster. cohesion separation
Internal Measures: Silhouette Coefficient • Silhouette Coefficient combine ideas of both cohesion and separation, but for individual points, as well as clusters and clusterings • For an individual point, i – Calculate a = average distance of i to the points in its cluster – Calculate b = min (average distance of i to points in another cluster) – The silhouette coefficient for a point is then given by s = 1 – a/b if a < b, (or s = b/a - 1 if a ≥ b, not the usual case) – Typically between 0 and 1. – The closer to 1 the better. • Can calculate the Average Silhouette width for a cluster or a clustering a b
External Measures of Cluster Validity: Entropy and Purity
Final Comment on Cluster Validity “The validation of clustering structures is the most difficult and frustrating part of cluster analysis. Without a strong effort in this direction, cluster analysis will remain a black art accessible only to those true believers who have experience and great courage.” Algorithms for Clustering Data, Jain and Dubes
Cluster Analysis • What is Cluster Analysis? • Types of Data in Cluster Analysis • Partitioning Methods • Hierarchical Methods • Density-Based Methods • Cluster Evaluation • Grid-Based Methods • Model-Based Clustering Methods • Outlier Analysis • Summary
Grid-Based Clustering Method • Using multi-resolution grid data structure • Several interesting methods – STING (a STatistical INformation Grid approach) by Wang, Yang and Muntz (1997) – WaveCluster by Sheikholeslami, Chatterjee, and Zhang (VLDB’98) • A multi-resolution clustering approach using wavelet method – CLIQUE: Agrawal, et al. (SIGMOD’98)
STING: A Statistical Information Grid Approach • Wang, Yang and Muntz (VLDB’97) • The spatial area area is divided into rectangular cells • There are several levels of cells corresponding to different levels of resolution
STING: A Statistical Information Grid Approach (2) – Each cell at a high level is partitioned into a number of smaller cells in the next lower level – Statistical info of each cell is calculated and stored beforehand and is used to answer queries – Parameters of higher level cells can be easily calculated from parameters of lower level cell • count, mean, s, min, max • type of distribution—normal, uniform, etc. – Use a top-down approach to answer spatial data queries – Start from a pre-selected layer—typically with a small number of cells – For each cell in the current level compute the confidence interval
STING: A Statistical Information Grid Approach (3) – Remove the irrelevant cells from further consideration – When finish examining the current layer, proceed to the next lower level – Repeat this process until the bottom layer is reached – Advantages: • Query-independent, easy to parallelize, incremental update • O(K), where K is the number of grid cells at the lowest level – Disadvantages: • All the cluster boundaries are either horizontal or vertical, and no diagonal boundary is detected
WaveCluster (1998) • Sheikholeslami, Chatterjee, and Zhang (VLDB’98) • A multi-resolution clustering approach which applies wavelet transform to the feature space – A wavelet transform is a signal processing technique that decomposes a signal into different frequency sub-band. • Both grid-based and density-based • Input parameters: – # of grid cells for each dimension – the wavelet, and the # of applications of wavelet transform.
What is Wavelet (1)?
WaveCluster (1998) • How to apply wavelet transform to find clusters – Summaries the data by imposing a multidimensional grid structure onto data space – These multidimensional spatial data objects are represented in a n-dimensional feature space – Apply wavelet transform on feature space to find the dense regions in the feature space – Apply wavelet transform multiple times which result in clusters at different scales from fine to coarse
Wavelet Transform • Decomposes a signal into different frequency subbands. (can be applied to n-dimensional signals) • Data are transformed to preserve relative distance between objects at different levels of resolution. • Allows natural clusters to become more distinguishable
What Is Wavelet (2)?
Quantization
Transformation
WaveCluster (1998) • Why is wavelet transformation useful for clustering – Unsupervised clustering It uses hat-shape filters to emphasize region where points cluster, but simultaneously to suppress weaker information in their boundary – Effective removal of outliers – Multi-resolution – Cost efficiency • Major features: – Complexity O(N) – Detect arbitrary shaped clusters at different scales – Not sensitive to noise, not sensitive to input order – Only applicable to low dimensional data
CLIQUE (Clustering In QUEst) • Agrawal, Gehrke, Gunopulos, Raghavan (SIGMOD’98). • Automatically identifying subspaces of a high dimensional data space that allow better clustering than original space • CLIQUE can be considered as both density-based and grid-based – It partitions each dimension into the same number of equal length interval – It partitions an m-dimensional data space into non-overlapping rectangular units – A unit is dense if the fraction of total data points contained in the unit exceeds the input model parameter – A cluster is a maximal set of connected dense units within a subspace
CLIQUE: The Major Steps • Partition the data space and find the number of points that lie inside each cell of the partition. • Identify the subspaces that contain clusters using the Apriori principle • Identify clusters: – Determine dense units in all subspaces of interests – Determine connected dense units in all subspaces of interests. • Generate minimal description for the clusters – Determine maximal regions that cover a cluster of connected dense units for each cluster – Determination of minimal cover for each cluster
Salary (10,000) 20 30 40 50 60 age 54312670 20 30 40 50 60 age 54312670 Vacation (week) age Vacation Salary 30 50 τ = 3
Strength and Weakness of CLIQUE • Strength – It automatically finds subspaces of the highest dimensionality such that high density clusters exist in those subspaces – It is insensitive to the order of records in input and does not presume some canonical data distribution – It scales linearly with the size of input and has good scalability as the number of dimensions in the data increases • Weakness – The accuracy of the clustering result may be degraded at the expense of simplicity of the method
Cluster Analysis • What is Cluster Analysis? • Types of Data in Cluster Analysis • Partitioning Methods • Hierarchical Methods • Density-Based Methods • Cluster Evaluation • Grid-Based Methods • Model-Based Clustering Methods • Outlier Analysis • Summary
Model-Based Clustering Methods • Attempt to optimize the fit between the data and some mathematical model • Statistical and AI approach – Conceptual clustering • A form of clustering in machine learning • Produces a classification scheme for a set of unlabeled objects • Finds characteristic description for each concept (class) – COBWEB (Fisher’87) • A popular a simple method of incremental conceptual learning • Creates a hierarchical clustering in the form of a classification tree • Each node refers to a concept and contains a probabilistic description of that concept
COBWEB Clustering Method A classification tree
More on Statistical-Based Clustering • Limitations of COBWEB – The assumption that the attributes are independent of each other is often too strong because correlation may exist – Not suitable for clustering large database data – skewed tree and expensive probability distributions • CLASSIT – an extension of COBWEB for incremental clustering of continuous data – suffers similar problems as COBWEB • AutoClass (Cheeseman and Stutz, 1996) – Uses Bayesian statistical analysis to estimate the number of clusters – Popular in industry
Other Model-Based Clustering Methods • Neural network approaches – Represent each cluster as an exemplar, acting as a “prototype” of the cluster – New objects are distributed to the cluster whose exemplar is the most similar according to some dostance measure • Competitive learning – Involves a hierarchical architecture of several units (neurons) – Neurons compete in a “winner-takes-all” fashion for the object currently being presented
Model-Based Clustering Methods
Self-organizing feature maps (SOMs) • Clustering is also performed by having several units competing for the current object • The unit whose weight vector is closest to the current object wins • The winner and its neighbors learn by having their weights adjusted • SOMs are believed to resemble processing that can occur in the brain • Useful for visualizing high-dimensional data in 2- or 3-D space
Cluster Analysis • What is Cluster Analysis? • Types of Data in Cluster Analysis • Partitioning Methods • Hierarchical Methods • Density-Based Methods • Cluster Evaluation • Grid-Based Methods • Model-Based Clustering Methods • Outlier Analysis • Summary
What Is Outlier Discovery? • What are outliers? – The set of objects are considerably dissimilar from the remainder of the data – Example: Sports: Michael Jordon, Wayne Gretzky, ... • Problem – Find top n outlier points • Applications: – Credit card fraud detection – Telecom fraud detection – Customer segmentation – Medical analysis
Outlier Discovery: Statistical Approaches Assume a model underlying distribution that generates data set (e.g. normal distribution) • Use discordancy tests depending on – data distribution – distribution parameter (e.g., mean, variance) – number of expected outliers • Drawbacks – most tests are for single attribute – In many cases, data distribution may not be known
Outlier Discovery: Distance- Based Approach • Introduced to counter the main limitations imposed by statistical methods – We need multi-dimensional analysis without knowing data distribution. • Distance-based outlier: A DB(p, D)-outlier is an object O in a dataset T such that at least a fraction p of the objects in T lies at a distance greater than D from O • Algorithms for mining distance-based outliers – Index-based algorithm – Nested-loop algorithm – Cell-based algorithm
Outlier Discovery: Deviation-Based Approach • Identifies outliers by examining the main characteristics of objects in a group • Objects that “deviate” from this description are considered outliers • sequential exception technique – simulates the way in which humans can distinguish unusual objects from among a series of supposedly like objects • OLAP data cube technique – uses data cubes to identify regions of anomalies in large multidimensional data
Cluster Analysis • What is Cluster Analysis? • Types of Data in Cluster Analysis • Partitioning Methods • Hierarchical Methods • Density-Based Methods • Cluster Evaluation • Grid-Based Methods • Model-Based Clustering Methods • Outlier Analysis • Summary
Problems and Challenges • Considerable progress has been made in scalable clustering methods – Partitioning: k-means, k-medoids, CLARANS – Hierarchical: BIRCH, CURE – Density-based: DBSCAN, CLIQUE, OPTICS – Grid-based: STING, WaveCluster – Model-based: Autoclass, Denclue, Cobweb • Current clustering techniques do not address all the requirements adequately • Constraint-based clustering analysis: Constraints exist in data space (bridges and highways) or in user queries
Constraint-Based Clustering Analysis • Clustering analysis: less parameters but more user- desired constraints, e.g., an ATM allocation problem
Clustering With Obstacle Objects Taking obstacles into account Not Taking obstacles into account
Summary • Cluster analysis groups objects based on their similarity and has wide applications • Measure of similarity can be computed for various types of data • Clustering algorithms can be categorized into partitioning methods, hierarchical methods, density-based methods, grid-based methods, and model-based methods • Outlier detection and analysis are very useful for fraud detection, etc. and can be performed by statistical, distance-based or deviation-based approaches • There are still lots of research issues on cluster analysis, such as constraint-based clustering
References (1) • R. Agrawal, J. Gehrke, D. Gunopulos, and P. Raghavan. Automatic subspace clustering of high dimensional data for data mining applications. SIGMOD'98 • M. R. Anderberg. Cluster Analysis for Applications. Academic Press, 1973. • M. Ankerst, M. Breunig, H.-P. Kriegel, and J. Sander. Optics: Ordering points to identify the clustering structure, SIGMOD’99. • P. Arabie, L. J. Hubert, and G. De Soete. Clustering and Classification. World Scietific, 1996 • M. Ester, H.-P. Kriegel, J. Sander, and X. Xu. A density-based algorithm for discovering clusters in large spatial databases. KDD'96. • M. Ester, H.-P. Kriegel, and X. Xu. Knowledge discovery in large spatial databases: Focusing techniques for efficient class identification. SSD'95. • D. Fisher. Knowledge acquisition via incremental conceptual clustering. Machine Learning, 2:139-172, 1987. • D. Gibson, J. Kleinberg, and P. Raghavan. Clustering categorical data: An approach based on dynamic systems. In Proc. VLDB’98. • S. Guha, R. Rastogi, and K. Shim. Cure: An efficient clustering algorithm for large databases. SIGMOD'98. • A. K. Jain and R. C. Dubes. Algorithms for Clustering Data. Printice Hall, 1988.
References (2) • L. Kaufman and P. J. Rousseeuw. Finding Groups in Data: an Introduction to Cluster Analysis. John Wiley & Sons, 1990. • E. Knorr and R. Ng. Algorithms for mining distance-based outliers in large datasets. VLDB’98. • G. J. McLachlan and K.E. Bkasford. Mixture Models: Inference and Applications to Clustering. John Wiley and Sons, 1988. • P. Michaud. Clustering techniques. Future Generation Computer systems, 13, 1997. • R. Ng and J. Han. Efficient and effective clustering method for spatial data mining. VLDB'94. • E. Schikuta. Grid clustering: An efficient hierarchical clustering method for very large data sets. Proc. 1996 Int. Conf. on Pattern Recognition, 101-105. • G. Sheikholeslami, S. Chatterjee, and A. Zhang. WaveCluster: A multi-resolution clustering approach for very large spatial databases. VLDB’98. • W. Wang, Yang, R. Muntz, STING: A Statistical Information grid Approach to Spatial Data Mining, VLDB’97. • T. Zhang, R. Ramakrishnan, and M. Livny. BIRCH : an efficient data clustering method for very large databases. SIGMOD'96.

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Clusteryanam

  • 1. A Brief Dealing of Clustering Prof. Dr. N.B. Venkateswarlu Aditya Institute of Technology and Management Tekkali Venkat_ritch@yahoo.com
  • 2. Cluster Analysis • What is Cluster Analysis? • Types of Data in Cluster Analysis • Partitioning Methods • Hierarchical Methods • Density-Based Methods • Cluster Evaluation • Grid-Based Methods • Model-Based Clustering Methods • Outlier Analysis • Summary
  • 3. The Problem of Clustering • Given a set of points, with a notion of distance between points, group the points into some number of clusters, so that members of a cluster are in some sense as close to each other as possible.
  • 4. What is Cluster Analysis? • Finding groups of objects such that the objects in a group will be similar (or related) to one another and different from (or unrelated to) the objects in other groups Inter-cluster distances are maximized Intra-cluster distances are minimized
  • 5. What is Cluster Analysis? • Cluster: a collection of data objects – Similar to one another within the same cluster – Dissimilar to the objects in other clusters • Cluster analysis – Grouping a set of data objects into clusters • Clustering is unsupervised classification: no predefined classes • Typical applications – As a stand-alone tool to get insight into data distribution – As a preprocessing step for other algorithms
  • 6. General Applications of Clustering • Pattern Recognition • Spatial Data Analysis – create thematic maps in GIS by clustering feature spaces – detect spatial clusters and explain them in spatial data mining • Image Processing • Economic Science (especially market research) • WWW – Document classification – Cluster Weblog data to discover groups of similar access patterns
  • 7. Examples of Clustering Applications • Marketing: Help marketers discover distinct groups in their customer bases, and then use this knowledge to develop targeted marketing programs • Land use: Identification of areas of similar land use in an earth observation database • Insurance: Identifying groups of motor insurance policy holders with a high average claim cost • City-planning: Identifying groups of houses according to their house type, value, and geographical location • Earth-quake studies: Observed earth quake epicenters should be clustered along continent faults
  • 8. Example: SkyCat • A catalog of 2 billion “sky objects” represented objects by their radiation in 9 dimensions (frequency bands). • Problem: cluster into similar objects, e.g., galaxies, nearby stars, quasars, etc. • Sloan Sky Survey is a newer, better version.
  • 9. Example: Clustering CD’s (Collaborative Filtering) • Intuitively: music divides into categories, and customers prefer a few categories. – But what are categories really? • Represent a CD by the customers who bought it. • Similar CD’s have similar sets of customers, and vice-versa.
  • 10. The Space of CD’s • Think of a space with one dimension for each customer. – Values in a dimension may be 0 or 1 only. • A CD’s point in this space is (x1, x2,…, xk), where xi = 1 iff the i th customer bought the CD. – Compare with the “correlated items” matrix: rows = customers; cols. = CD’s.
  • 11. Example: Clustering Documents • Represent a document by a vector (x1, x2,…, xk), where xi = 1 iff the i th word (in some order) appears in the document. – It actually doesn’t matter if k is infinite; i.e., we don’t limit the set of words. • Documents with similar sets of words may be about the same topic.
  • 12. Example: Protein Sequences • Objects are sequences of {C,A,T,G}. • Distance between sequences is edit distance, the minimum number of inserts and deletes needed to turn one into the other. • Note there is a “distance,” but no convenient space in which points “live.”
  • 13. Notion of a Cluster can be Ambiguous How many clusters? Four ClustersTwo Clusters Six Clusters
  • 14. Problems With Clustering • Clustering in two dimensions looks easy. • Clustering small amounts of data looks easy. • And in most cases, looks are not deceiving.
  • 15. The Curse of Dimensionality • Many applications involve not 2, but 10 or 10,000 dimensions. • High-dimensional spaces look different: almost all pairs of points are at about the same distance. – Assuming random points within a bounding box, e.g., values between 0 and 1 in each dimension.
  • 16. What Is Good Clustering? • A good clustering method will produce high quality clusters with – high intra-class similarity – low inter-class similarity • The quality of a clustering result depends on both the similarity measure used by the method and its implementation. • The quality of a clustering method is also measured by its ability to discover some or all of the hidden patterns.
  • 17. Types of Clusterings • A clustering is a set of clusters • Important distinction between hierarchical and partitional sets of clusters • Partitional Clustering – A division data objects into non-overlapping subsets (clusters) such that each data object is in exactly one subset • Hierarchical clustering – A set of nested clusters organized as a hierarchical tree
  • 18. Partitional Clustering Original Points A Partitional Clustering
  • 19. Hierarchical Clustering p4 p1 p3 p2 p4p1 p2 p3 p4p1 p2 p3 Traditional Hierarchical Clustering Non-traditional Hierarchical Clustering Non-traditional Dendrogram Traditional Dendrogram
  • 20. Other Distinctions Between Sets of Clusters • Exclusive versus non-exclusive – In non-exclusive clusterings, points may belong to multiple clusters. – Can represent multiple classes or ‘border’ points • Fuzzy versus non-fuzzy – In fuzzy clustering, a point belongs to every cluster with some weight between 0 and 1 – Weights must sum to 1 – Probabilistic clustering has similar characteristics • Partial versus complete – In some cases, we only want to cluster some of the data • Heterogeneous versus homogeneous – Cluster of widely different sizes, shapes, and densities
  • 21. Types of Clusters • Well-separated clusters • Center-based clusters • Contiguous clusters • Density-based clusters • Property or Conceptual • Described by an Objective Function
  • 22. Types of Clusters: Well- Separated • Well-Separated Clusters: – A cluster is a set of points such that any point in a cluster is closer (or more similar) to every other point in the cluster than to any point not in the cluster. 3 well-separated clusters
  • 23. Types of Clusters: Center- Based • Center-based – A cluster is a set of objects such that an object in a cluster is closer (more similar) to the “center” of a cluster, than to the center of any other cluster – The center of a cluster is often a centroid, the average of all the points in the cluster, or a medoid, the most “representative” point of a cluster 4 center-based clusters
  • 24. Types of Clusters: Contiguity- Based • Contiguous Cluster (Nearest neighbor or Transitive) – A cluster is a set of points such that a point in a cluster is closer (or more similar) to one or more other points in the cluster than to any point not in the cluster. 8 contiguous clusters
  • 25. Types of Clusters: Density- Based • Density-based – A cluster is a dense region of points, which is separated by low- density regions, from other regions of high density. – Used when the clusters are irregular or intertwined, and when noise and outliers are present. 6 density-based clusters
  • 26. Types of Clusters: Conceptual Clusters • Shared Property or Conceptual Clusters – Finds clusters that share some common property or represent a particular concept. 2 Overlapping Circles
  • 27. Types of Clusters: Objective Function • Clusters Defined by an Objective Function – Finds clusters that minimize or maximize an objective function. – Enumerate all possible ways of dividing the points into clusters and evaluate the `goodness' of each potential set of clusters by using the given objective function. (NP Hard) – Can have global or local objectives. • Hierarchical clustering algorithms typically have local objectives • Partitional algorithms typically have global objectives – A variation of the global objective function approach is to fit the data to a parameterized model. • Parameters for the model are determined from the data. • Mixture models assume that the data is a ‘mixture' of a number of statistical distributions.
  • 28. Types of Clusters: Objective Function … • Map the clustering problem to a different domain and solve a related problem in that domain – Proximity matrix defines a weighted graph, where the nodes are the points being clustered, and the weighted edges represent the proximities between points – Clustering is equivalent to breaking the graph into connected components, one for each cluster. – Want to minimize the edge weight between clusters and maximize the edge weight within clusters
  • 29. Requirements of Clustering in Data Mining • Scalability • Ability to deal with different types of attributes • Discovery of clusters with arbitrary shape • Minimal requirements for domain knowledge to determine input parameters • Able to deal with noise and outliers • Insensitive to order of input records • High dimensionality • Incorporation of user-specified constraints • Interpretability and usability
  • 30. Data Structures • Data matrix – (two modes) • Dissimilarity matrix – (one mode)                   npx...nfx...n1x ............... ipx...ifx...i1x ............... 1px...1fx...11x                 0...)2,()1,( ::: )2,3() ...ndnd 0dd(3,1 0d(2,1) 0
  • 31. Measure the Quality of Clustering • Dissimilarity/Similarity metric: Similarity is expressed in terms of a distance function, which is typically metric: d(i, j) • There is a separate “quality” function that measures the “goodness” of a cluster. • The definitions of distance functions are usually very different for interval-scaled, boolean, categorical, ordinal and ratio variables. • Weights should be associated with different variables based on applications and data semantics. • It is hard to define “similar enough” or “good enough” – the answer is typically highly subjective.
  • 32. Type of data in clustering analysis • Interval-scaled variables: • Binary variables: • Nominal, ordinal, and ratio variables: • Variables of mixed types:
  • 33. Interval-valued variables • Standardize data – Calculate the mean absolute deviation: where – Calculate the standardized measurement (z-score) • Using mean absolute deviation is more robust than using standard deviation .)... 21 1 nffff xx(xnm +++= |)|...|||(|1 21 fnffffff mxmxmxns −++−+−= f fif if s mx z − =
  • 34. Similarity and Dissimilarity Between Objects • Distances are normally used to measure the similarity or dissimilarity between two data objects • Some popular ones include: Minkowski distance: where i = (xi1, xi2, …, xip) and j = (xj1, xj2, …, xjp) are two p-dimensional data objects, and q is a positive integer • If q = 1, d is Manhattan distance q q pp qq j x i x j x i x j x i xjid )||...|||(|),( 2211 −++−+−= ||...||||),( 2211 pp j x i x j x i x j x i xjid −++−+−=
  • 35. Similarity and Dissimilarity Between Objects (Cont.) • If q = 2, d is Euclidean distance: – Properties • d(i,j) ≥ 0 • d(i,i) = 0 • d(i,j) = d(j,i) • d(i,j) ≤ d(i,k) + d(k,j) • Also, one can use weighted distance, parametric Pearson product moment correlation, or other disimilarity measures )||...|||(|),( 22 22 2 11 pp j x i x j x i x j x i xjid −++−+−=
  • 36. Distance Measures • Each clustering problem is based on some kind of “distance” between points. • Two major classes of distance measure: 1. Euclidean 2. Non-Euclidean
  • 37. Euclidean Vs. Non- Euclidean • A Euclidean space has some number of real-valued dimensions and “dense” points. – There is a notion of “average” of two points. – A Euclidean distance is based on the locations of points in such a space. • A Non-Euclidean distance is based on properties of points, but not their “location” in a space.
  • 38. Axioms of a Distance Measure • d is a distance measure if it is a function from pairs of points to reals such that: 1. d(x,y) > 0. 2. d(x,y) = 0 iff x = y. 3. d(x,y) = d(y,x). 4. d(x,y) < d(x,z) + d(z,y) (triangle inequality ).
  • 39. Some Euclidean Distances • L2 norm : d(x,y) = square root of the sum of the squares of the differences between x and y in each dimension. – The most common notion of “distance.” • L1 norm : sum of the differences in each dimension. – Manhattan distance = distance if you had to travel along coordinates only.
  • 40. Examples of Euclidean Distances x = (5,5) y = (9,8) L2-norm: dist(x,y) = √(42 +32 ) = 5 L1-norm: dist(x,y) = 4+3 = 7 4 35
  • 41. Another Euclidean Distance • L∞ norm : d(x,y) = the maximum of the differences between x and y in any dimension. • Note: the maximum is the limit as n goes to ∞ of what you get by taking the n th power of the differences, summing and taking the n th root.
  • 42. Non-Euclidean Distances • Jaccard distance for sets = 1 minus ratio of sizes of intersection and union. • Cosine distance = angle between vectors from the origin to the points in question. • Edit distance = number of inserts and deletes to change one string into another.
  • 43. Jaccard Distance • Example: p1 = 10111; p2 = 10011. – Size of intersection = 3; size of union = 4, Jaccard measure (not distance) = 3/4. • Need to make a distance function satisfying triangle inequality and other laws. • d(x,y) = 1 – (Jaccard measure) works.
  • 44. Why J.D. Is a Distance Measure • d(x,x) = 0 because x∩x = x∪x. • d(x,y) = d(y,x) because union and intersection are symmetric. • d(x,y) > 0 because |x∩y| < |x∪y|. • d(x,y) < d(x,z) + d(z,y) trickier --- next slide.
  • 45. Triangle Inequality for J.D. 1 - |x ∩z| + 1 - |y ∩z| > 1 - |x ∩y| |x ∪z| |y ∪z| |x ∪y| • Remember: |a ∩b|/|a ∪b| = probability that minhash(a) = minhash(b). • Thus, 1 - |a ∩b|/|a ∪b| = probability that minhash(a) ≠ minhash(b).
  • 46. Triangle Inequality --- (2) • So we need to observe that prob[minhash(x) ≠ minhash(y)] < prob[minhash(x) ≠ minhash(z)] + prob[minhash(z) ≠ minhash(y)] • Clincher: whenever minhash(x) ≠ minhash(y), one of minhash(x) ≠ minhash(z) and minhash(z) ≠ minhash(y) must be true.
  • 47. Cosine Distance • Think of a point as a vector from the origin (0,0,…,0) to its location. • Two points’ vectors make an angle, whose cosine is the normalized dot-product of the vectors: p1.p2/|p2||p1|. – Example p1 = 00111; p2 = 10011. – p1.p2 = 2; |p1| = |p2| = √3. – cos(θ) = 2/3; θ is about 48 degrees.
  • 49. Why? p1 = (x1,y1) p2 = (x2,0)x1 θ Dot product is invariant under rotation, so pick convenient coordinate system. x1 = p1.p2/|p2| p1.p2 = x1*x2. |p2| = x2.
  • 50. Why C.D. Is a Distance Measure • d(x,x) = 0 because arccos(1) = 0. • d(x,y) = d(y,x) by symmetry. • d(x,y) > 0 because angles are chosen to be in the range 0 to 180 degrees. • Triangle inequality: physical reasoning. If I rotate an angle from x to z and then from z to y, I can’t rotate less than from x to y.
  • 51. Edit Distance • The edit distance of two strings is the number of inserts and deletes of characters needed to turn one into the other. • Equivalently, d(x,y) = |x| + | y| -2|LCS(x,y)|. – LCS = longest common subsequence = longest string obtained both by deleting from x and deleting from y.
  • 52. Example • x = abcde ; y = bcduve. • Turn x into y by deleting a, then inserting u and v after d. – Edit-distance = 3. • Or, LCS(x,y) = bcde. • |x| + |y| - 2|LCS(x,y)| = 5 + 6 –2*4 = 3.
  • 53. Why E.D. Is a Distance Measure • d(x,x) = 0 because 0 edits suffice. • d(x,y) = d(y,x) because insert/delete are inverses of each other. • d(x,y) > 0: no notion of negative edits. • Triangle inequality: changing x to z and then to y is one way to change x to y.
  • 54. Variant Edit Distance • Allow insert, delete, and mutate. – Change one character into another. • Minimum number of inserts, deletes, and mutates also forms a distance measure.
  • 55. Binary Variables • A contingency table for binary data • Simple matching coefficient (invariant, if the binary variable is symmetric): • Jaccard coefficient (noninvariant if the binary variable is asymmetric): dcba cbjid +++ +=),( pdbcasum dcdc baba sum ++ + + 0 1 01 cba cbjid ++ +=),( Object i Object j
  • 56. Dissimilarity between Binary Variables • Example – gender is a symmetric attribute – the remaining attributes are asymmetric binary – let the values Y and P be set to 1, and the value N be set to 0 Name Gender Fever Cough Test-1 Test-2 Test-3 Test-4 Jack M Y N P N N N Mary F Y N P N P N Jim M Y P N N N N 75.0 211 21 ),( 67.0 111 11 ),( 33.0 102 10 ),( = ++ + = = ++ + = = ++ + = maryjimd jimjackd maryjackd
  • 57. Nominal Variables • A generalization of the binary variable in that it can take more than 2 states, e.g., red, yellow, blue, green • Method 1: Simple matching – m: # of matches, p: total # of variables • Method 2: use a large number of binary variables – creating a new binary variable for each of the M nominal states p mpjid −=),(
  • 58. Ordinal Variables • An ordinal variable can be discrete or continuous • Order is important, e.g., rank • Can be treated like interval-scaled – replace xif by their rank – map the range of each variable onto [0, 1] by replacing i-th object in the f-th variable by – compute the dissimilarity using methods for interval-scaled variables 1 1 − − = f if if M r z },...,1{ fif Mr ∈
  • 59. Ratio-Scaled Variables • Ratio-scaled variable: a positive measurement on a nonlinear scale, approximately at exponential scale, such as AeBt or Ae-Bt • Methods: – treat them like interval-scaled variables—not a good choice! (why?—the scale can be distorted) – apply logarithmic transformation yif = log(xif) – treat them as continuous ordinal data treat their rank as interval- scaled
  • 60. Variables of Mixed Types • A database may contain all the six types of variables – symmetric binary, asymmetric binary, nominal, ordinal, interval and ratio • One may use a weighted formula to combine their effects – f is binary or nominal: dij (f) = 0 if xif = xjf , or dij (f) = 1 o.w. – f is interval-based: use the normalized distance – f is ordinal or ratio-scaled • compute ranks rif and • and treat zif as interval-scaled )( 1 )()( 1 ),( f ij p f f ij f ij p f d jid δ δ = = Σ Σ = 1 1 − − = f if M rzif
  • 61. Cluster Analysis • What is Cluster Analysis? • Types of Data in Cluster Analysis • Partitioning Methods • Hierarchical Methods • Density-Based Methods • Cluster Evaluation • Grid-Based Methods • Model-Based Clustering Methods • Outlier Analysis • Summary
  • 62. Partitioning Algorithms: Basic Concept • Partitioning method: Construct a partition of a database D of n objects into a set of k clusters • Given a k, find a partition of k clusters that optimizes the chosen partitioning criterion – Global optimal: exhaustively enumerate all partitions – Heuristic methods: k-means and k-medoids algorithms – k-means (MacQueen’67): Each cluster is represented by the center of the cluster – k-medoids or PAM (Partition around medoids) (Kaufman & Rousseeuw’87): Each cluster is represented by one of the objects in the cluster
  • 63. The K-Means Clustering Method • Given k, the k-means algorithm is implemented in four steps: – Partition objects into k nonempty subsets – Compute seed points as the centroids of the clusters of the current partition (the centroid is the center, i.e., mean point, of the cluster) – Assign each object to the cluster with the nearest seed point – Go back to Step 2, stop when no more new assignment
  • 64. The K-Means Clustering Method 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 K=2 Arbitrarily choose K object as initial cluster center Assign each objects to most similar center Update the cluster means Update the cluster means reassignreassign
  • 65. k –Means Algorithm(s) • Assumes Euclidean space. • Start by picking k, the number of clusters. • Initialize clusters by picking one point per cluster. – For instance, pick one point at random, then k -1 other points, each as far away as possible from the previous points.
  • 66. Populating Clusters 1. For each point, place it in the cluster whose current centroid it is nearest. 2. After all points are assigned, fix the centroids of the k clusters. 3. Reassign all points to their closest centroid. – Sometimes moves points between clusters.
  • 67. Example 1 2 3 4 5 6 7 8x x Clusters after first round Reassigned points
  • 68. Getting k Right – Try different k, looking at the change in the average distance to centroid, as k increases. • Average falls rapidly until right k, then changes little. k Average distance to centroid Best value of k
  • 69. Example x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x x x Too few; many long distances to centroid.
  • 70. Example x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x x x Just right; distances rather short.
  • 71. Example x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x x x Too many; little improvement in average distance.
  • 72. BFR Algorithm • BFR (Bradley-Fayyad-Reina) is a variant of k -means designed to handle very large (disk-resident) data sets. • It assumes that clusters are normally distributed around a centroid in a Euclidean space. – Standard deviations in different dimensions may vary.
  • 73. BFR --- (2) • Points are read one main-memory-full at a time. • Most points from previous memory loads are summarized by simple statistics. • To begin, from the initial load we select the initial k centroids by some sensible approach.
  • 74. Initialization: k -Means • Possibilities include: 1. Take a small sample and cluster optimally. 2. Take a sample; pick a random point, and then k – 1 more points, each as far from the previously selected points as possible.
  • 75. Three Classes of Points 1. The discard set : points close enough to a centroid to be represented statistically. 2. The compression set : groups of points that are close together but not close to any centroid. They are represented statistically, but not assigned to a cluster. 3. The retained set : isolated points.
  • 76. Representing Sets of Points • For each cluster, the discard set is represented by: 1. The number of points, N. 2. The vector SUM, whose i th component is the sum of the coordinates of the points in the i th dimension. 3. The vector SUMSQ: i th component = sum of squares of coordinates in i th dimension.
  • 77. Comments • 2d + 1 values represent any number of points. – d = number of dimensions. • Averages in each dimension (centroid coordinates) can be calculated easily as SUMi /N. – SUMi = i th component of SUM.
  • 78. Comments --- (2) • Variance of a cluster’s discard set in dimension i can be computed by: (SUMSQi /N ) – (SUMi /N )2 • And the standard deviation is the square root of that. • The same statistics can represent any compression set.
  • 79. “Galaxies” Picture A cluster. Its points are in the DS. The centroid Compressed sets. Their points are in the CS. Points in the RS
  • 80. Processing a “Memory- Load” of Points 1. Find those points that are “sufficiently close” to a cluster centroid; add those points to that cluster and the DS. 2. Use any main-memory clustering algorithm to cluster the remaining points and the old RS. Clusters go to the CS; outlying points to the RS.
  • 81. Processing --- (2) 3. Adjust statistics of the clusters to account for the new points. 4. Consider merging compressed sets in the CS. 5. If this is the last round, merge all compressed sets in the CS and all RS points into their nearest cluster.
  • 82. A Few Details . . . • How do we decide if a point is “close enough” to a cluster that we will add the point to that cluster? • How do we decide whether two compressed sets deserve to be combined into one?
  • 83. How Close is Close Enough? • We need a way to decide whether to put a new point into a cluster. • BFR suggest two ways: 1. The Mahalanobis distance is less than a threshold. 2. Low likelihood of the currently nearest centroid changing.
  • 84. Mahalanobis Distance • Normalized Euclidean distance. • For point (x1,…,xk) and centroid (c1,…,ck): 1. Normalize in each dimension: yi = |xi -ci|/σi 2. Take sum of the squares of the yi ’s. 3. Take the square root.
  • 85. Mahalanobis Distance --- (2) • If clusters are normally distributed in d dimensions, then one standard deviation corresponds to a distance √d. – I.e., 70% of the points of the cluster will have a Mahalanobis distance < √d. • Accept a point for a cluster if its M.D. is < some threshold, e.g. 4 standard deviations.
  • 86. Picture: Equal M.D. Regions σ 2σ
  • 87. Should Two CS Subclusters Be Combined? • Compute the variance of the combined subcluster. – N, SUM, and SUMSQ allow us to make that calculation. • Combine if the variance is below some threshold.
  • 88. Comments on the K-Means Method • Strength: Relatively efficient: O(tkn), where n is # objects, k is # clusters, and t is # iterations. Normally, k, t << n. • Comparing: PAM: O(k(n-k)2 ), CLARA: O(ks2 + k(n-k)) • Comment: Often terminates at a local optimum. The global optimum may be found using techniques such as: deterministic annealing and genetic algorithms • Weakness – Applicable only when mean is defined, then what about categorical data? – Need to specify k, the number of clusters, in advance – Unable to handle noisy data and outliers – Not suitable to discover clusters with non-convex shapes
  • 89. Variations of the K-Means Method • A few variants of the k-means which differ in – Selection of the initial k means – Dissimilarity calculations – Strategies to calculate cluster means • Handling categorical data: k-modes (Huang’98) – Replacing means of clusters with modes – Using new dissimilarity measures to deal with categorical objects – Using a frequency-based method to update modes of clusters – A mixture of categorical and numerical data: k-prototype method
  • 90. What is the problem of k- Means Method? • The k-means algorithm is sensitive to outliers ! – Since an object with an extremely large value may substantially distort the distribution of the data. • K-Medoids: Instead of taking the mean value of the object in a cluster as a reference point, medoids can be used, which is the most centrally located object in a cluster. 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10
  • 91. Importance of Choosing Initial Centroids -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 0.5 1 1.5 2 2.5 3 x y Iteration 1 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 0.5 1 1.5 2 2.5 3 x y Iteration 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 0.5 1 1.5 2 2.5 3 x y Iteration 3 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 0.5 1 1.5 2 2.5 3 x y Iteration 4 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 0.5 1 1.5 2 2.5 3 x y Iteration 5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 0.5 1 1.5 2 2.5 3 x y Iteration 6
  • 92. Solutions to Initial Centroids Problem • Multiple runs – Helps, but probability is not on your side • Sample and use hierarchical clustering to determine initial centroids • Select more than k initial centroids and then select among these initial centroids – Select most widely separated • Postprocessing • Bisecting K-means – Not as susceptible to initialization issues
  • 93. Limitations of K-means: Differing Density Original Points K-means (3 Clusters)
  • 94. Limitations of K-means: Non- globular Shapes Original Points K-means (2 Clusters)
  • 95. The CURE Algorithm • Problem with BFR/k -means: – Assumes clusters are normally distributed in each dimension. – And axes are fixed --- ellipses at an angle are not OK. • CURE: – Assumes a Euclidean distance. – Allows clusters to assume any shape.
  • 96. Example: Stanford Faculty Salaries e e e e e e e e e e e h h h h h h h h h h h h h salary age
  • 97. Starting CURE 1. Pick a random sample of points that fit in main memory. 2. Cluster these points hierarchically --- group nearest points/clusters. 3. For each cluster, pick a sample of points, as dispersed as possible. 4. From the sample, pick representatives by moving them (say) 20% toward the centroid of the cluster.
  • 98. Example: Initial Clusters e e e e e e e e e e e h h h h h h h h h h h h h salary age
  • 99. Example: Pick Dispersed Points e e e e e e e e e e e h h h h h h h h h h h h h salary age Pick (say) 4 remote points for each cluster.
  • 100. Example: Pick Dispersed Points e e e e e e e e e e e h h h h h h h h h h h h h salary age Move points (say) 20% toward the centroid.
  • 101. Finishing CURE • Now, visit each point p in the data set. • Place it in the “closest cluster.” – Normal definition of “closest”: that cluster with the closest (to p ) among all the sample points of all the clusters.
  • 102. Curse of Dimensionality • One way to look at it: in large-dimension spaces, random vectors are perpendicular. Why? Argument #1: Lots of 2-dim subspaces. There must be one where the vectors’ projections are almost perpendicular. Argument #2: Expected value of cosine of angle is 0.
  • 103. Cosine of Angle Between Random Vectors • Assume vectors emanate from the origin (0,0,…,0). • Components are random in range [-1,1]. • (a1,a2,…,an).(b1,b2,…,bn) has expected value 0 and a standard deviation that grows as √n. • But lengths of both vectors grow as √n. • So dot product around √n/ (√n * √n) = 1/√n.
  • 104. Random Vectors --- Continued • Thus, a typical pair of vectors has an angle whose cosine is on the order of 1/√n. • As n -> ∞, that’s 0; i.e., the angle is about 90°.
  • 105. Interesting Consequence • Suppose “random vectors are perpendicular,” even in non-Euclidean spaces. • Suppose we know the distance from A to B, say d (A,B ), and we also know d (B,C ), but we don’t know d (A,C ). • Suppose B and C are fairly close, say in the same cluster. • What is d (A,C )?
  • 106. Diagram of Situation A C B Approximately perpendicular Assuming points lie in a plane: d (A,B )2 + d (B,C )2 = d (A,C )2
  • 107. Important Point • Why do we assume AB is perpendicular to AC, and not that either of the other two angles are right- angles? 1. AB and AC are not “random vectors”; they each go to points that are far away from A and close to each other. 2. If AB is longer than AC, then it is angle ACB that is right, but both ACB and ABC are approximately right-angles.
  • 108. Dealing With a Non- Euclidean Space • Problem: clusters cannot be represented by centroids. • Why? Because the “average” of “points” might not be a point in the space. • Best substitute: the clustroid = point in the cluster that minimizes the sum of the squares of distances to the points in the cluster.
  • 109. Representing Clusters in Non- Euclidean Spaces • Recall BFR represents a Euclidean cluster by N, SUM, and SUMSQ. • A non-Euclidean cluster is represented by: – N. – The clustroid. – Sum of the squares of the distances from clustroid to all points in the cluster.
  • 110. Example of CoD Use • Problem: in non-Euclidean space, we want to decide whether to merge two clusters. – Each cluster represented by N, clustroid, and “SUMSQ.” – Also, SUMSQ for each point in the cluster, even if it is not the clustroid. • Merge if SUMSQ for new cluster is “low.”
  • 111. Estimating SUMSQ p p ’s clustroid, c other clust- roid, b
  • 112. Suppose p Were the Clustroid of Combined Cluster • It’s SUMSQ would be the sum of: 1. Old SUMSQ(p) [for old cluster containing p]. 2. SUMSQ(b) plus d (p,b)2 times number of points in b ’s cluster. • Critical point: vector p ->b assumed perpendicular to vectors from b to all other points in its cluster --- justifies (2).
  • 113. Combining Clusters --- Continued • We can thus estimate SUMSQ for each point in the combined cluster. Take the point with the least SUMSQ as the clustroid of the new cluster --- provided that SUMSQ is small enough.
  • 114. The GRGPF Algorithm • From Ganti et al. --- see reading list. • Works for non-Euclidean distances. • Works for massive (disk-resident) data. • Hierarchical clustering. • Clusters are grouped into a tree of disk blocks (like a B-tree or R-tree).
  • 115. Information Retained About a Cluster 1. N, clustroid, SUMSQ. 2. The p points closest to the clustroid, and their values of SUMSQ. 3. The p points of the cluster that are furthest away from the clustroid, and their SUMSQ’s.
  • 116. At Interior Nodes of the Tree • Interior nodes have samples of the clustroids of the clusters found at descendant leaves of this node. • Try to keep clusters on one leaf block close, descendants of a level-1 node close, etc. • Interior part of tree kept in main memory.
  • 117. Picture of the Tree cluster data cluster data samples main memory on disk
  • 118. Initialization • Take a main-memory sample of points. • Organize them into clusters hierarchically. • Build the initial tree, with level-1 interior nodes representing clusters of clusters, and so on. • All other points are inserted into this tree.
  • 119. Inserting Points • Start at the root. • At each interior node, visit one or more children that have sample clustroids near the inserted point. • At the leaves, insert the point into the cluster with the nearest clustroid.
  • 120. Updating Cluster Data • Suppose we add point X to a cluster. • Increase count N by 1. • For each of the 2p + 1 points Y whose SUMSQ is stored, add d (X,Y )2 . • Estimate SUMSQ for X.
  • 121. Estimating SUMSQ(X ) • If C is the clustroid, SUMSQ(X ) is, by the CoD assumption: Nd (X,C ) 2 + SUMSQ(C ) – Based on assumption that vector from X to C is perpendicular to vectors from C to all the other nodes of the cluster. • This value may allow X to replace one of the closest or furthest nodes.
  • 122. Possible Modification to Cluster Data • There may be a new clustroid --- one of the p closest points --- because of the addition of X. • Eventually, the clustroid may migrate out of the p closest points, and the entire representation of the cluster needs to be recomputed.
  • 123. Splitting and Merging Clusters • Maintain a threshold for the radius of a cluster = √(SUMSQ/N ). • Split a cluster whose radius is too large. • Adding clusters may overflow leaf blocks, and require splits of blocks up the tree. – Splitting is similar to a B-tree. – But try to keep locality of clusters.
  • 124. Splitting and Merging --- (2) • The problem case is when we have split so much that the tree no longer fits in main memory. • Raise the threshold on radius and merge clusters that are sufficiently close.
  • 125. Merging Clusters • Suppose there are nearby clusters with clustroids C and D, and we want to consider merging them. • Assume that the clustroid of the combined cluster will be one of the p furthest points from the clustroid of one of those clusters.
  • 126. Merging --- (2) • Compute SUMSQ(X ) [from the cluster of C ] for the combined cluster by summing: 1. SUMSQ(X ) from its own cluster. 2. SUMSQ(D ) + N [d (X,C )2 + d (C,D )2 ]. Uses the CoD to reason that the distance from X to each point in the other cluster goes to C, makes a right angle to D, and another right angle to the point.
  • 127. Merging --- Concluded • Pick as the clustroid for the combined cluster that point with the least SUMSQ. • But if this SUMSQ is too large, do not merge clusters. • Hope you get enough mergers to fit the tree in main memory.
  • 128. Fastmap • Not a clustering algorithm --- rather, a method for applying multidimensional scaling. – That is, mapping the points onto a small- dimension space, so the CoD does not apply.
  • 129. Fastmap --- (2) • Assumes non-Euclidean space. – But like GRGFP pretends it is working in 2- dimensional Euclidean space when it is convenient to do so. • Goal: map n points in much less than O(n 2 ) time. – I.e., you cannot compute distances between each pair of points and place points in k-dim. space to minimize error.
  • 130. Fastmap --- Key Idea • Create a “dimension” in non-Euclidean space by: 1. Pick a pair of points A and B that are far apart. Start with random A; pick most distant B. 2. Treat AB as an “axis” and project all points onto AB, using the law of cosines.
  • 131. Projecting Point C Onto AB A B C x d(A,C) d(B,C) d(A,B) x = [d 2 (A,C) + d 2 (A,B) – d 2 (B,C)]/2d (A,B)
  • 132. Revising Distances • Having computed the position of every point along the pseudo-axis AB, we need to lower the distances between points in the “other dimensions.”
  • 134. But … • We can’t afford to compute new distances for each pseudo-dimension. – It would take O(n 2 ) time. • Rather, for each pseudo-dimension, store the position along the pseudo-axis for each point, and adjust the distance between points by square-subtract-sqrt only when needed. – I.e., one of the points is an axis-end.
  • 135. Fastmap --- Summary • Pick a number of dimensions k. FOR i = 1 TO k DO BEGIN Pick a pseudo-axis AiBi; Compute projection of each point onto this pseudo-axis; END; • Each step is O(ni ); total O(nk 2 ).
  • 136. The CURE Algorithm • Problem with BFR/k -means: – Assumes clusters are normally distributed in each dimension. – And axes are fixed --- ellipses at an angle are not OK. • CURE: – Assumes a Euclidean distance. – Allows clusters to assume any shape.
  • 137. Example: Stanford Faculty Salaries e e e e e e e e e e e h h h h h h h h h h h h h salary age
  • 138. Starting CURE 1. Pick a random sample of points that fit in main memory. 2. Cluster these points hierarchically --- group nearest points/clusters. 3. For each cluster, pick a sample of points, as dispersed as possible. 4. From the sample, pick representatives by moving them (say) 20% toward the centroid of the cluster.
  • 139. Example: Initial Clusters e e e e e e e e e e e h h h h h h h h h h h h h salary age
  • 140. Example: Pick Dispersed Points e e e e e e e e e e e h h h h h h h h h h h h h salary age Pick (say) 4 remote points for each cluster.
  • 141. Example: Pick Dispersed Points e e e e e e e e e e e h h h h h h h h h h h h h salary age Move points (say) 20% toward the centroid.
  • 142. Finishing CURE • Now, visit each point p in the data set. • Place it in the “closest cluster.” – Normal definition of “closest”: that cluster with the closest (to p ) among all the sample points of all the clusters.
  • 143. Clustering a Stream (New Topic) • Assume points enter in a stream. • Maintain a sliding window of points. • Queries ask for clusters of points within some suffix of the window. • Only important issue: where are the cluster centroids. – There is no notion of “all the points” in a stream.
  • 144. BDMO Approach • BDMO = Babcock, Datar, Motwani, O’Callaghan. • k –means based. • Can use less than O(N ) space for windows of size N. • Generalizes trick of DGIM: buckets of increasing “weight.”
  • 145. Recall DGIM • Maintains a sequence of buckets B1, B2, … • Buckets have timestamps (most recent stream element in bucket). • Sizes of buckets nondecreasing. – In DGIM size = power of 2. • Either 1 or 2 of each size.
  • 146. Alternative Combining Rule • Instead of “combine the 2nd and 3rd of any one size” we could say: • “Combine Bi+1 and Bi if size(Bi+1 ∪ Bi) < size(Bi-1 ∪ Bi-2 ∪ … ∪ B1).” – If Bi+1, Bi, and Bi-1 are the same size, inequality must hold (almost). – If Bi-1 is smaller, it cannot hold.
  • 147. Buckets for Clustering • In place of “size” (number of 1’s) we use (an approximation to) the sum of the distances from all points to the centroid of their cluster. • Merge consecutive buckets if the “size” of the merged bucket is less than the sum of the sizes of all later buckets.
  • 148. Consequence of Merge Rule • In a stable list of buckets, any two consecutive buckets are “bigger” than all smaller buckets. • Thus, “sizes” grow exponentially. • If there is a limit on total “size,” then the number of buckets is O(log N ). • N = window size. – E.g., all points are in a fixed hypercube.
  • 149. Outline of Algorithm 1. What do buckets look like? – Clusters at various levels, represented by centroids. 2. How do we merge buckets? – Keep # of clusters at each level small. 3. What happens when we query? – Final clustering of all clusters of all relevant buckets.
  • 150. Organization of Buckets • Each bucket consists of clusters at some number of levels. – 4 levels in our examples. • Clusters represented by: 1. Location of centroid. 2. Weight = number of points in the cluster. 3. Cost = upper bound on sum of distances from member points to centroid.
  • 151. Processing Buckets --- (1) • Actions determined by N (window size) and k (desired number of clusters). • Also uses a tuning parameter τ for which we use 1/4 to simplify. – 1/τ is the number of levels of clusters.
  • 152. Processing Buckets --- (2) • Initialize a new bucket with k new points. – Each is a cluster at level 0. • If the timestamp of the oldest bucket is outside the window, delete that bucket.
  • 153. Level-0 Clusters • A single point p is represented by (p, 1, 0). • That is: 1. A point is its own centroid. 2. The cluster has one point. 3. The sum of distances to the centroid is 0.
  • 154. Merging Buckets --- (1) • Needed in two situations: 1. We have to process a query, which requires us to (temporarily) merge some tail of the bucket sequence. 2. We have just added a new (most recent) bucket and we need to check the rule about two consecutive buckets being “bigger” than all that follow.
  • 155. Merging Buckets --- (2) • Step 1: Take the union of the clusters at each level. • Step 2: If the number of clusters (points) at level 0 is now more than N 1/4 , cluster them into k clusters. – These become clusters at level 1. • Steps 3,…: Repeat, going up the levels, if needed.
  • 156. Representing New Clusters • Centroid = weighted average of centroids of component clusters. • Weight = sum of weights. • Cost = sum over all component clusters of: 1. Cost of component cluster. 2. Weight of component times distance from its centroid to new centroid.
  • 157. Example: New Centroid + (18,-2) + (3,3) + (12,12) centroids 5 10 15 weights + (12,2) new centroid
  • 158. Example: New Costs + (18,-2) + (3,3) + (12,12) 5 10 15 + (12,2) old cost added true cost
  • 159. Queries • Find all the buckets within the range of the query. – The last bucket may be only partially within the range. • Cluster all clusters at all levels into k clusters. • Return the k centroids.
  • 160. Error in Estimation • Goal is to pick the k centroids that minimize the true cost (sum of distances from each point to its centroid). • Since recorded “costs” are inexact, there can be a factor of 2 error at each level. • Additional error because some of last bucket may not belong. – But fraction of spurious points is small (why?).
  • 161. Effect of Cost-Errors 1. Alter when buckets get combined. Not really important. 2. Produce suboptimal clustering at any stage of the algorithm. The real measure of how bad the output is.
  • 162. Speedup of Algorithm • As given, algorithm is slow. – Each new bucket causes O(log N ) bucket- merger problems. • A faster version allows the first bucket to have not k, but N 1/2 (or in general N 2τ ) points. – A number of consequences, including slower queries, more space.
  • 163. The K-Medoids Clustering Method • Find representative objects, called medoids, in clusters • PAM (Partitioning Around Medoids, 1987) – starts from an initial set of medoids and iteratively replaces one of the medoids by one of the non-medoids if it improves the total distance of the resulting clustering – PAM works effectively for small data sets, but does not scale well for large data sets • CLARA (Kaufmann & Rousseeuw, 1990) • CLARANS (Ng & Han, 1994): Randomized sampling • Focusing + spatial data structure (Ester et al., 1995)
  • 164. Typical k-medoids algorithm (PAM) 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Total Cost = 20 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 K=2 Arbitrary choose k object as initial medoids 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Assign each remainin g object to nearest medoids Randomly select a nonmedoid object,Oramdom Compute total cost of swapping 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Total Cost = 26 Swapping O and Oramdom If quality is improved. Do loop Until no change 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10
  • 165. PAM (Partitioning Around Medoids) (1987) • PAM (Kaufman and Rousseeuw, 1987), built in Splus • Use real object to represent the cluster – Select k representative objects arbitrarily – For each pair of non-selected object h and selected object i, calculate the total swapping cost TCih – For each pair of i and h, • If TCih < 0, i is replaced by h • Then assign each non-selected object to the most similar representative object – repeat steps 2-3 until there is no change
  • 166. PAM Clustering: Total swapping cost TCih=∑jCjih 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 j i h t Cjih = 0 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 t i h j Cjih = d(j, h) - d(j, i) 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 h i t j Cjih = d(j, t) - d(j, i) 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 t i h j Cjih = d(j, h) - d(j, t)
  • 167. What is the problem with PAM? • Pam is more robust than k-means in the presence of noise and outliers because a medoid is less influenced by outliers or other extreme values than a mean • Pam works efficiently for small data sets but does not scale well for large data sets. – O(k(n-k)2 ) for each iteration where n is # of data,k is # of clusters Sampling based method, CLARA(Clustering LARge Applications)
  • 168. CLARA (Clustering Large Applications) (1990) • CLARA (Kaufmann and Rousseeuw in 1990) – Built in statistical analysis packages, such as S+ • It draws multiple samples of the data set, applies PAM on each sample, and gives the best clustering as the output • Strength: deals with larger data sets than PAM • Weakness: – Efficiency depends on the sample size – A good clustering based on samples will not necessarily represent a good clustering of the whole data set if the sample is biased
  • 169. CLARANS (“Randomized” CLARA) (1994) • CLARANS (A Clustering Algorithm based on Randomized Search) (Ng and Han’94) • CLARANS draws sample of neighbors dynamically • The clustering process can be presented as searching a graph where every node is a potential solution, that is, a set of k medoids • If the local optimum is found, CLARANS starts with new randomly selected node in search for a new local optimum • It is more efficient and scalable than both PAM and CLARA • Focusing techniques and spatial access structures may further improve its performance (Ester et al.’95)
  • 170. Cluster Analysis • What is Cluster Analysis? • Types of Data in Cluster Analysis • Partitioning Methods • Hierarchical Methods • Density-Based Methods • Cluster Evaluation • Grid-Based Methods • Model-Based Clustering Methods • Outlier Analysis • Summary
  • 171. Hierarchical Clustering • Use distance matrix as clustering criteria. This method does not require the number of clusters k as an input, but needs a termination condition Step 0 Step 1 Step 2 Step 3 Step 4 b d c e a a b d e c d e a b c d e Step 4 Step 3 Step 2 Step 1 Step 0 agglomerative (AGNES) divisive (DIANA)
  • 172. AGNES (Agglomerative Nesting) • Introduced in Kaufmann and Rousseeuw (1990) • Implemented in statistical analysis packages, e.g., Splus • Use the Single-Link method and the dissimilarity matrix. • Merge nodes that have the least dissimilarity • Go on in a non-descending fashion • Eventually all nodes belong to the same cluster 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10
  • 173. Agglomerative Clustering Algorithm • More popular hierarchical clustering technique • Basic algorithm is straightforward 1. Compute the proximity matrix 2. Let each data point be a cluster 3. Repeat 4. Merge the two closest clusters 5. Update the proximity matrix 6. Until only a single cluster remains • Key operation is the computation of the proximity of two clusters – Different approaches to defining the distance between clusters distinguish the different algorithms
  • 174. Starting Situation • Start with clusters of individual points and a proximity matrix p1 p3 p5 p4 p2 p1 p2 p3 p4 p5 . . . . . . Proximity Matrix
  • 175. Intermediate Situation • After some merging steps, we have some clusters C1 C4 C2 C5 C3 C2C1 C1 C3 C5 C4 C2 C3 C4 C5 Proximity Matrix
  • 176. Intermediate Situation • We want to merge the two closest clusters (C2 and C5) and update the proximity matrix. C1 C4 C2 C5 C3 C2C1 C1 C3 C5 C4 C2 C3 C4 C5 Proximity Matrix
  • 177. After Merging • The question is “How do we update the proximity matrix?” C1 C4 C2 U C5 C3 ? ? ? ? ? ? ? C2 U C5C1 C1 C3 C4 C2 U C5 C3 C4 Proximity Matrix
  • 178. How to Define Inter-Cluster Similarity p1 p3 p5 p4 p2 p1 p2 p3 p4 p5 . . . . . . Similarity? • MIN • MAX • Group Average • Distance Between Centroids • Other methods driven by an objective function – Ward’s Method uses squared error Proximity Matrix
  • 179. How to Define Inter-Cluster Similarity p1 p3 p5 p4 p2 p1 p2 p3 p4 p5 . . . . . . Proximity Matrix • MIN • MAX • Group Average • Distance Between Centroids • Other methods driven by an objective function – Ward’s Method uses squared error
  • 180. How to Define Inter-Cluster Similarity p1 p3 p5 p4 p2 p1 p2 p3 p4 p5 . . . . . . Proximity Matrix • MIN • MAX • Group Average • Distance Between Centroids • Other methods driven by an objective function – Ward’s Method uses squared error
  • 181. How to Define Inter-Cluster Similarity p1 p3 p5 p4 p2 p1 p2 p3 p4 p5 . . . . . . Proximity Matrix • MIN • MAX • Group Average • Distance Between Centroids • Other methods driven by an objective function – Ward’s Method uses squared error
  • 182. How to Define Inter-Cluster Similarity p1 p3 p5 p4 p2 p1 p2 p3 p4 p5 . . . . . . Proximity Matrix • MIN • MAX • Group Average • Distance Between Centroids • Other methods driven by an objective function – Ward’s Method uses squared error × ×
  • 183. Hierarchical Clustering: Time and Space requirements • O(N2 ) space since it uses the proximity matrix. – N is the number of points. • O(N3 ) time in many cases – There are N steps and at each step the size, N2 , proximity matrix must be updated and searched – Complexity can be reduced to O(N2 log(N) ) time for some approaches
  • 184. Hierarchical Clustering: Problems and Limitations • Once a decision is made to combine two clusters, it cannot be undone • No objective function is directly minimized • Different schemes have problems with one or more of the following: – Sensitivity to noise and outliers – Difficulty handling different sized clusters and convex shapes – Breaking large clusters
  • 185. A Dendrogram Shows How the Clusters are Merged Hierarchically • Decompose data objects into a several levels of nested partitioning (tree of clusters), called a dendrogram. • A clustering of the data objects is obtained by cutting the dendrogram at the desired level, then each connected component forms a cluster.
  • 186. DIANA (Divisive Analysis) • Introduced in Kaufmann and Rousseeuw (1990) • Implemented in statistical analysis packages, e.g., Splus • Inverse order of AGNES • Eventually each node forms a cluster on its own 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10
  • 187. More on Hierarchical Clustering Methods • Major weakness of agglomerative clustering methods – do not scale well: time complexity of at least O(n2 ), where n is the number of total objects – can never undo what was done previously • Integration of hierarchical with distance-based clustering – BIRCH (1996): uses CF-tree and incrementally adjusts the quality of sub-clusters – CURE (1998): selects well-scattered points from the cluster and then shrinks them towards the center of the cluster by a specified fraction – CHAMELEON (1999): hierarchical clustering using dynamic modeling
  • 188. BIRCH (1996) • Birch: Balanced Iterative Reducing and Clustering using Hierarchies, by Zhang, Ramakrishnan, Livny (SIGMOD’96) • Incrementally construct a CF (Clustering Feature) tree, a hierarchical data structure for multiphase clustering – Phase 1: scan DB to build an initial in-memory CF tree (a multi-level compression of the data that tries to preserve the inherent clustering structure of the data) – Phase 2: use an arbitrary clustering algorithm to cluster the leaf nodes of the CF-tree • Scales linearly: finds a good clustering with a single scan and improves the quality with a few additional scans • Weakness: handles only numeric data, and sensitive to the order of the data record.
  • 189. Clustering Feature: CF = (N, LS, SS) N: Number of data points LS: ∑N i=1=Xi SS: ∑N i=1=Xi 2 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 CF = (5, (16,30),(54,190)) (3,4) (2,6) (4,5) (4,7) (3,8) Clustering Feature Vector
  • 190. CF-Tree in BIRCH • Clustering feature: – summary of the statistics for a given subcluster: the 0-th, 1st and 2nd moments of the subcluster from the statistical point of view. – registers crucial measurements for computing cluster and utilizes storage efficiently A CF tree is a height-balanced tree that stores the clustering features for a hierarchical clustering – A nonleaf node in a tree has descendants or “children” – The nonleaf nodes store sums of the CFs of their children • A CF tree has two parameters – Branching factor: specify the maximum number of children. – threshold: max diameter of sub-clusters stored at the leaf nodes
  • 191. CF Tree CF1 child1 CF3 child3 CF2 child2 CF6 child6 CF1 child1 CF3 child3 CF2 child2 CF5 child5 CF1 CF2 CF6 prev next CF1 CF2 CF4 prev next B = 7 L = 6 Root Non-leaf node Leaf node Leaf node
  • 192. CURE (Clustering Using REpresentatives ) • CURE: proposed by Guha, Rastogi & Shim, 1998 – Stops the creation of a cluster hierarchy if a level consists of k clusters – Uses multiple representative points to evaluate the distance between clusters, adjusts well to arbitrary shaped clusters and avoids single-link effect
  • 193. Drawbacks of Distance- Based Method • Drawbacks of square-error based clustering method – Consider only one point as representative of a cluster – Good only for convex shaped, similar size and density, and if k can be reasonably estimated
  • 194. Cure: The Algorithm • Draw random sample s. • Partition sample to p partitions with size s/p • Partially cluster partitions into s/pq clusters • Eliminate outliers – By random sampling – If a cluster grows too slow, eliminate it. • Cluster partial clusters. • Label data in disk
  • 195. Data Partitioning and Clustering – s = 50 – p = 2 – s/p = 25 x x x y y y y x y x s/pq = 5
  • 196. Cure: Shrinking Representative Points • Shrink the multiple representative points towards the gravity center by a fraction of α. • Multiple representatives capture the shape of the cluster x y x y
  • 197. Clustering Categorical Data: ROCK • ROCK: Robust Clustering using linKs, by S. Guha, R. Rastogi, K. Shim (ICDE’99). – Use links to measure similarity/proximity – Not distance based – Computational complexity: • Basic ideas: – Similarity function and neighbors: Let T1 = {1,2,3}, T2={3,4,5} O n nm m n nm a( log )2 2 + + Sim T T T T T T ( , )1 2 1 2 1 2 = ∩ ∪ Sim T T( , ) { } { , , , , } .1 2 3 1 2 3 4 5 1 5 0 2= = =
  • 198. CHAMELEON (Hierarchical clustering using dynamic modeling) • CHAMELEON: by G. Karypis, E.H. Han, and V. Kumar’99 • Measures the similarity based on a dynamic model – Two clusters are merged only if the interconnectivity and closeness (proximity) between two clusters are high relative to the internal interconnectivity of the clusters and closeness of items within the clusters – Cure ignores information about interconnectivity of the objects, Rock ignores information about the closeness of two clusters • A two-phase algorithm 1. Use a graph partitioning algorithm: cluster objects into a large number of relatively small sub-clusters 2. Use an agglomerative hierarchical clustering algorithm: find the genuine clusters by repeatedly combining these sub-clusters
  • 199. Overall Framework of CHAMELEON Construct Sparse Graph Partition the Graph Merge Partition Final Clusters Data Set
  • 200. Cluster Analysis • What is Cluster Analysis? • Types of Data in Cluster Analysis • Partitioning Methods • Hierarchical Methods • Density-Based Methods • Cluster Evaluation • Grid-Based Methods • Model-Based Clustering Methods • Outlier Analysis • Summary
  • 201. Density-Based Clustering Methods • Clustering based on density (local cluster criterion), such as density-connected points • Major features: – Discover clusters of arbitrary shape – Handle noise – One scan – Need density parameters as termination condition • Several interesting studies: – DBSCAN: Ester, et al. (KDD’96) – OPTICS: Ankerst, et al (SIGMOD’99). – DENCLUE: Hinneburg & D. Keim (KDD’98) – CLIQUE: Agrawal, et al. (SIGMOD’98)
  • 202. Density Concepts • Core object (CO)–object with at least ‘M’ objects within a radius ‘E-neighborhood’ • Directly density reachable (DDR)–x is CO, y is in x’s ‘E- neighborhood’ • Density reachable–there exists a chain of DDR objects from x to y • Density based cluster–density connected objects maximum w.r.t. reachability
  • 203. Density-Based Clustering: Background • Two parameters: – Eps: Maximum radius of the neighbourhood – MinPts: Minimum number of points in an Eps-neighbourhood of that point • NEps(p): {q belongs to D | dist(p,q) <= Eps} • Directly density-reachable: A point p is directly density- reachable from a point q wrt. Eps, MinPts if – 1) p belongs to NEps(q) – 2) core point condition: |NEps (q)| >= MinPts p q MinPts = 5 Eps = 1 cm
  • 204. Density-Based Clustering: Background (II) • Density-reachable: – A point p is density-reachable from a point q wrt. Eps, MinPts if there is a chain of points p1, …, pn, p1 = q, pn = p such that pi+1 is directly density- reachable from pi • Density-connected – A point p is density-connected to a point q wrt. Eps, MinPts if there is a point o such that both, p and q are density-reachable from o wrt. Eps and MinPts. p q p1 p q o
  • 205. DBSCAN: Density Based Spatial Clustering of Applications with Noise • Relies on a density-based notion of cluster: A cluster is defined as a maximal set of density-connected points • Discovers clusters of arbitrary shape in spatial databases with noise Core Border Outlier Eps = 1cm MinPts = 5
  • 206. DBSCAN • DBSCAN is a density-based algorithm. – Density = number of points within a specified radius (Eps) – A point is a core point if it has more than a specified number of points (MinPts) within Eps • These are points that are at the interior of a cluster – A border point has fewer than MinPts within Eps, but is in the neighborhood of a core point – A noise point is any point that is not a core point or a border point.
  • 207. DBSCAN: Core, Border, and Noise Points
  • 208. DBSCAN Algorithm • Eliminate noise points • Perform clustering on the remaining points
  • 209. DBSCAN: Core, Border and Noise Points Original Points Point types: core, border and noise Eps = 10, MinPts = 4
  • 210. When DBSCAN Works Well Original Points Clusters • Resistant to Noise • Can handle clusters of different shapes and sizes
  • 211. When DBSCAN Does NOT Work Well Original Points (MinPts=4, Eps=9.75). (MinPts=4, Eps=9.92) • Varying densities • High-dimensional data
  • 212. DBSCAN: Determining EPS and MinPts • Idea is that for points in a cluster, their kth nearest neighbors are at roughly the same distance • Noise points have the kth nearest neighbor at farther distance • So, plot sorted distance of every point to its kth nearest neighbor
  • 213. OPTICS: A Cluster-Ordering Method (1999) • OPTICS: Ordering Points To Identify the Clustering Structure – Ankerst, Breunig, Kriegel, and Sander (SIGMOD’99) – Produces a special order of the database wrt its density-based clustering structure – This cluster-ordering contains info equiv to the density-based clusterings corresponding to a broad range of parameter settings – Good for both automatic and interactive cluster analysis, including finding intrinsic clustering structure – Can be represented graphically or using visualization techniques
  • 214. Cluster Analysis • What is Cluster Analysis? • Types of Data in Cluster Analysis • Partitioning Methods • Hierarchical Methods • Density-Based Methods • Cluster Evaluation • Grid-Based Methods • Model-Based Clustering Methods • Outlier Analysis • Summary
  • 215. Cluster Validity • For supervised classification we have a variety of measures to evaluate how good our model is – Accuracy, precision, recall • For cluster analysis, the analogous question is how to evaluate the “goodness” of the resulting clusters? • But “clusters are in the eye of the beholder”! • Then why do we want to evaluate them? – To avoid finding patterns in noise – To compare clustering algorithms – To compare two sets of clusters – To compare two clusters
  • 216. Clusters found in Random Data 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x y Random Points 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x y K-means 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x y DBSCAN 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x y Complete Link
  • 217. Different Aspects of Cluster Validation 1. Determining the clustering tendency of a set of data, i.e., distinguishing whether non-random structure actually exists in the data. 2. Comparing the results of a cluster analysis to externally known results, e.g., to externally given class labels. 3. Evaluating how well the results of a cluster analysis fit the data without reference to external information. - Use only the data 4. Comparing the results of two different sets of cluster analyses to determine which is better. 5. Determining the ‘correct’ number of clusters. For 2, 3, and 4, we can further distinguish whether we want to evaluate the entire clustering or just individual clusters.
  • 218. Measures of Cluster Validity • Numerical measures that are applied to judge various aspects of cluster validity, are classified into the following three types. – External Index: Used to measure the extent to which cluster labels match externally supplied class labels. • Entropy – Internal Index: Used to measure the goodness of a clustering structure without respect to external information. • Sum of Squared Error (SSE) – Relative Index: Used to compare two different clusterings or clusters. • Often an external or internal index is used for this function, e.g., SSE or entropy • Sometimes these are referred to as criteria instead of indices – However, sometimes criterion is the general strategy and index is the numerical measure that implements the criterion.
  • 219. Measuring Cluster Validity Via Correlation • Two matrices – Proximity Matrix – “Incidence” Matrix • One row and one column for each data point • An entry is 1 if the associated pair of points belong to the same cluster • An entry is 0 if the associated pair of points belongs to different clusters • Compute the correlation between the two matrices – Since the matrices are symmetric, only the correlation between n(n-1) / 2 entries needs to be calculated. • High correlation indicates that points that belong to the same cluster are close to each other. • Not a good measure for some density or contiguity based clusters.
  • 220. Measuring Cluster Validity Via Correlation • Correlation of incidence and proximity matrices for the K-means clusterings of the following two data sets. 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x y 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x y Corr = -0.9235 Corr = -0.5810
  • 221. Using Similarity Matrix for Cluster Validation • Order the similarity matrix with respect to cluster labels and inspect visually. 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x y Points Points 20 40 60 80 100 10 20 30 40 50 60 70 80 90 100 Similarity 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
  • 222. Using Similarity Matrix for Cluster Validation • Clusters in random data are not so crisp Points Points 20 40 60 80 100 10 20 30 40 50 60 70 80 90 100 Similarity 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 DBSCAN 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x y
  • 223. Points Points 20 40 60 80 100 10 20 30 40 50 60 70 80 90 100 Similarity 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Using Similarity Matrix for Cluster Validation • Clusters in random data are not so crisp K-means 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 xy
  • 224. Using Similarity Matrix for Cluster Validation 1 2 3 5 6 4 7 DBSCAN 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 500 1000 1500 2000 2500 3000 500 1000 1500 2000 2500 3000
  • 225. Internal Measures: SSE • Clusters in more complicated figures aren’t well separated • Internal Index: Used to measure the goodness of a clustering structure without respect to external information – SSE • SSE is good for comparing two clusterings or two clusters (average SSE). • Can also be used to estimate the number of clusters 2 5 10 15 20 25 30 0 1 2 3 4 5 6 7 8 9 10 K SSE 5 10 15 -6 -4 -2 0 2 4 6
  • 226. Internal Measures: SSE • SSE curve for a more complicated data set 1 2 3 5 6 4 7 SSE of clusters found using K-means
  • 227. Framework for Cluster Validity • Need a framework to interpret any measure. – For example, if our measure of evaluation has the value, 10, is that good, fair, or poor? • Statistics provide a framework for cluster validity – The more “atypical” a clustering result is, the more likely it represents valid structure in the data – Can compare the values of an index that result from random data or clusterings to those of a clustering result. • If the value of the index is unlikely, then the cluster results are valid – These approaches are more complicated and harder to understand. • For comparing the results of two different sets of cluster analyses, a framework is less necessary. – However, there is the question of whether the difference between two index values is significant
  • 228. Statistical Framework for SSE • Example – Compare SSE of 0.005 against three clusters in random data – Histogram shows SSE of three clusters in 500 sets of random data points of size 100 distributed over the range 0.2 – 0.8 for x and y values 0.016 0.018 0.02 0.022 0.024 0.026 0.028 0.03 0.032 0.034 0 5 10 15 20 25 30 35 40 45 50 SSE Count 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x y
  • 229. Statistical Framework for Correlation • Correlation of incidence and proximity matrices for the K-means clusterings of the following two data sets. 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x y 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x y Corr = -0.9235 Corr = -0.5810
  • 230. Internal Measures: Cohesion and Separation • Cluster Cohesion: Measures how closely related are objects in a cluster – Example: SSE • Cluster Separation: Measure how distinct or well-separated a cluster is from other clusters • Example: Squared Error – Cohesion is measured by the within cluster sum of squares (SSE) – Separation is measured by the between cluster sum of squares – Where |Ci| is the size of cluster i ∑ ∑ ∈ −= i Cx i i mxWSS 2 )( ∑ −= i ii mmCBSS 2 )(
  • 231. Internal Measures: Cohesion and Separation • Example: SSE – BSS + WSS = constant 1 2 3 4 5 × ×× m1 m2 m 1091 9)35.4(2)5.13(2 1)5.45()5.44()5.12()5.11( 22 2222 =+= =−×+−×= =−+−+−+−= Total BSS WSSK=2 clusters: 10010 0)33(4 10)35()34()32()31( 2 2222 =+= =−×= =−+−+−+−= Total BSS WSSK=1 cluster:
  • 232. Internal Measures: Cohesion and Separation • A proximity graph based approach can also be used for cohesion and separation. – Cluster cohesion is the sum of the weight of all links within a cluster. – Cluster separation is the sum of the weights between nodes in the cluster and nodes outside the cluster. cohesion separation
  • 233. Internal Measures: Silhouette Coefficient • Silhouette Coefficient combine ideas of both cohesion and separation, but for individual points, as well as clusters and clusterings • For an individual point, i – Calculate a = average distance of i to the points in its cluster – Calculate b = min (average distance of i to points in another cluster) – The silhouette coefficient for a point is then given by s = 1 – a/b if a < b, (or s = b/a - 1 if a ≥ b, not the usual case) – Typically between 0 and 1. – The closer to 1 the better. • Can calculate the Average Silhouette width for a cluster or a clustering a b
  • 234. External Measures of Cluster Validity: Entropy and Purity
  • 235. Final Comment on Cluster Validity “The validation of clustering structures is the most difficult and frustrating part of cluster analysis. Without a strong effort in this direction, cluster analysis will remain a black art accessible only to those true believers who have experience and great courage.” Algorithms for Clustering Data, Jain and Dubes
  • 236. Cluster Analysis • What is Cluster Analysis? • Types of Data in Cluster Analysis • Partitioning Methods • Hierarchical Methods • Density-Based Methods • Cluster Evaluation • Grid-Based Methods • Model-Based Clustering Methods • Outlier Analysis • Summary
  • 237. Grid-Based Clustering Method • Using multi-resolution grid data structure • Several interesting methods – STING (a STatistical INformation Grid approach) by Wang, Yang and Muntz (1997) – WaveCluster by Sheikholeslami, Chatterjee, and Zhang (VLDB’98) • A multi-resolution clustering approach using wavelet method – CLIQUE: Agrawal, et al. (SIGMOD’98)
  • 238. STING: A Statistical Information Grid Approach • Wang, Yang and Muntz (VLDB’97) • The spatial area area is divided into rectangular cells • There are several levels of cells corresponding to different levels of resolution
  • 239. STING: A Statistical Information Grid Approach (2) – Each cell at a high level is partitioned into a number of smaller cells in the next lower level – Statistical info of each cell is calculated and stored beforehand and is used to answer queries – Parameters of higher level cells can be easily calculated from parameters of lower level cell • count, mean, s, min, max • type of distribution—normal, uniform, etc. – Use a top-down approach to answer spatial data queries – Start from a pre-selected layer—typically with a small number of cells – For each cell in the current level compute the confidence interval
  • 240. STING: A Statistical Information Grid Approach (3) – Remove the irrelevant cells from further consideration – When finish examining the current layer, proceed to the next lower level – Repeat this process until the bottom layer is reached – Advantages: • Query-independent, easy to parallelize, incremental update • O(K), where K is the number of grid cells at the lowest level – Disadvantages: • All the cluster boundaries are either horizontal or vertical, and no diagonal boundary is detected
  • 241. WaveCluster (1998) • Sheikholeslami, Chatterjee, and Zhang (VLDB’98) • A multi-resolution clustering approach which applies wavelet transform to the feature space – A wavelet transform is a signal processing technique that decomposes a signal into different frequency sub-band. • Both grid-based and density-based • Input parameters: – # of grid cells for each dimension – the wavelet, and the # of applications of wavelet transform.
  • 243. WaveCluster (1998) • How to apply wavelet transform to find clusters – Summaries the data by imposing a multidimensional grid structure onto data space – These multidimensional spatial data objects are represented in a n-dimensional feature space – Apply wavelet transform on feature space to find the dense regions in the feature space – Apply wavelet transform multiple times which result in clusters at different scales from fine to coarse
  • 244. Wavelet Transform • Decomposes a signal into different frequency subbands. (can be applied to n-dimensional signals) • Data are transformed to preserve relative distance between objects at different levels of resolution. • Allows natural clusters to become more distinguishable
  • 248. WaveCluster (1998) • Why is wavelet transformation useful for clustering – Unsupervised clustering It uses hat-shape filters to emphasize region where points cluster, but simultaneously to suppress weaker information in their boundary – Effective removal of outliers – Multi-resolution – Cost efficiency • Major features: – Complexity O(N) – Detect arbitrary shaped clusters at different scales – Not sensitive to noise, not sensitive to input order – Only applicable to low dimensional data
  • 249. CLIQUE (Clustering In QUEst) • Agrawal, Gehrke, Gunopulos, Raghavan (SIGMOD’98). • Automatically identifying subspaces of a high dimensional data space that allow better clustering than original space • CLIQUE can be considered as both density-based and grid-based – It partitions each dimension into the same number of equal length interval – It partitions an m-dimensional data space into non-overlapping rectangular units – A unit is dense if the fraction of total data points contained in the unit exceeds the input model parameter – A cluster is a maximal set of connected dense units within a subspace
  • 250. CLIQUE: The Major Steps • Partition the data space and find the number of points that lie inside each cell of the partition. • Identify the subspaces that contain clusters using the Apriori principle • Identify clusters: – Determine dense units in all subspaces of interests – Determine connected dense units in all subspaces of interests. • Generate minimal description for the clusters – Determine maximal regions that cover a cluster of connected dense units for each cluster – Determination of minimal cover for each cluster
  • 251. Salary (10,000) 20 30 40 50 60 age 54312670 20 30 40 50 60 age 54312670 Vacation (week) age Vacation Salary 30 50 τ = 3
  • 252. Strength and Weakness of CLIQUE • Strength – It automatically finds subspaces of the highest dimensionality such that high density clusters exist in those subspaces – It is insensitive to the order of records in input and does not presume some canonical data distribution – It scales linearly with the size of input and has good scalability as the number of dimensions in the data increases • Weakness – The accuracy of the clustering result may be degraded at the expense of simplicity of the method
  • 253. Cluster Analysis • What is Cluster Analysis? • Types of Data in Cluster Analysis • Partitioning Methods • Hierarchical Methods • Density-Based Methods • Cluster Evaluation • Grid-Based Methods • Model-Based Clustering Methods • Outlier Analysis • Summary
  • 254. Model-Based Clustering Methods • Attempt to optimize the fit between the data and some mathematical model • Statistical and AI approach – Conceptual clustering • A form of clustering in machine learning • Produces a classification scheme for a set of unlabeled objects • Finds characteristic description for each concept (class) – COBWEB (Fisher’87) • A popular a simple method of incremental conceptual learning • Creates a hierarchical clustering in the form of a classification tree • Each node refers to a concept and contains a probabilistic description of that concept
  • 256. More on Statistical-Based Clustering • Limitations of COBWEB – The assumption that the attributes are independent of each other is often too strong because correlation may exist – Not suitable for clustering large database data – skewed tree and expensive probability distributions • CLASSIT – an extension of COBWEB for incremental clustering of continuous data – suffers similar problems as COBWEB • AutoClass (Cheeseman and Stutz, 1996) – Uses Bayesian statistical analysis to estimate the number of clusters – Popular in industry
  • 257. Other Model-Based Clustering Methods • Neural network approaches – Represent each cluster as an exemplar, acting as a “prototype” of the cluster – New objects are distributed to the cluster whose exemplar is the most similar according to some dostance measure • Competitive learning – Involves a hierarchical architecture of several units (neurons) – Neurons compete in a “winner-takes-all” fashion for the object currently being presented
  • 259. Self-organizing feature maps (SOMs) • Clustering is also performed by having several units competing for the current object • The unit whose weight vector is closest to the current object wins • The winner and its neighbors learn by having their weights adjusted • SOMs are believed to resemble processing that can occur in the brain • Useful for visualizing high-dimensional data in 2- or 3-D space
  • 260. Cluster Analysis • What is Cluster Analysis? • Types of Data in Cluster Analysis • Partitioning Methods • Hierarchical Methods • Density-Based Methods • Cluster Evaluation • Grid-Based Methods • Model-Based Clustering Methods • Outlier Analysis • Summary
  • 261. What Is Outlier Discovery? • What are outliers? – The set of objects are considerably dissimilar from the remainder of the data – Example: Sports: Michael Jordon, Wayne Gretzky, ... • Problem – Find top n outlier points • Applications: – Credit card fraud detection – Telecom fraud detection – Customer segmentation – Medical analysis
  • 262. Outlier Discovery: Statistical Approaches Assume a model underlying distribution that generates data set (e.g. normal distribution) • Use discordancy tests depending on – data distribution – distribution parameter (e.g., mean, variance) – number of expected outliers • Drawbacks – most tests are for single attribute – In many cases, data distribution may not be known
  • 263. Outlier Discovery: Distance- Based Approach • Introduced to counter the main limitations imposed by statistical methods – We need multi-dimensional analysis without knowing data distribution. • Distance-based outlier: A DB(p, D)-outlier is an object O in a dataset T such that at least a fraction p of the objects in T lies at a distance greater than D from O • Algorithms for mining distance-based outliers – Index-based algorithm – Nested-loop algorithm – Cell-based algorithm
  • 264. Outlier Discovery: Deviation-Based Approach • Identifies outliers by examining the main characteristics of objects in a group • Objects that “deviate” from this description are considered outliers • sequential exception technique – simulates the way in which humans can distinguish unusual objects from among a series of supposedly like objects • OLAP data cube technique – uses data cubes to identify regions of anomalies in large multidimensional data
  • 265. Cluster Analysis • What is Cluster Analysis? • Types of Data in Cluster Analysis • Partitioning Methods • Hierarchical Methods • Density-Based Methods • Cluster Evaluation • Grid-Based Methods • Model-Based Clustering Methods • Outlier Analysis • Summary
  • 266. Problems and Challenges • Considerable progress has been made in scalable clustering methods – Partitioning: k-means, k-medoids, CLARANS – Hierarchical: BIRCH, CURE – Density-based: DBSCAN, CLIQUE, OPTICS – Grid-based: STING, WaveCluster – Model-based: Autoclass, Denclue, Cobweb • Current clustering techniques do not address all the requirements adequately • Constraint-based clustering analysis: Constraints exist in data space (bridges and highways) or in user queries
  • 267. Constraint-Based Clustering Analysis • Clustering analysis: less parameters but more user- desired constraints, e.g., an ATM allocation problem
  • 268. Clustering With Obstacle Objects Taking obstacles into account Not Taking obstacles into account
  • 269. Summary • Cluster analysis groups objects based on their similarity and has wide applications • Measure of similarity can be computed for various types of data • Clustering algorithms can be categorized into partitioning methods, hierarchical methods, density-based methods, grid-based methods, and model-based methods • Outlier detection and analysis are very useful for fraud detection, etc. and can be performed by statistical, distance-based or deviation-based approaches • There are still lots of research issues on cluster analysis, such as constraint-based clustering
  • 270. References (1) • R. Agrawal, J. Gehrke, D. Gunopulos, and P. Raghavan. Automatic subspace clustering of high dimensional data for data mining applications. SIGMOD'98 • M. R. Anderberg. Cluster Analysis for Applications. Academic Press, 1973. • M. Ankerst, M. Breunig, H.-P. Kriegel, and J. Sander. Optics: Ordering points to identify the clustering structure, SIGMOD’99. • P. Arabie, L. J. Hubert, and G. De Soete. Clustering and Classification. World Scietific, 1996 • M. Ester, H.-P. Kriegel, J. Sander, and X. Xu. A density-based algorithm for discovering clusters in large spatial databases. KDD'96. • M. Ester, H.-P. Kriegel, and X. Xu. Knowledge discovery in large spatial databases: Focusing techniques for efficient class identification. SSD'95. • D. Fisher. Knowledge acquisition via incremental conceptual clustering. Machine Learning, 2:139-172, 1987. • D. Gibson, J. Kleinberg, and P. Raghavan. Clustering categorical data: An approach based on dynamic systems. In Proc. VLDB’98. • S. Guha, R. Rastogi, and K. Shim. Cure: An efficient clustering algorithm for large databases. SIGMOD'98. • A. K. Jain and R. C. Dubes. Algorithms for Clustering Data. Printice Hall, 1988.
  • 271. References (2) • L. Kaufman and P. J. Rousseeuw. Finding Groups in Data: an Introduction to Cluster Analysis. John Wiley & Sons, 1990. • E. Knorr and R. Ng. Algorithms for mining distance-based outliers in large datasets. VLDB’98. • G. J. McLachlan and K.E. Bkasford. Mixture Models: Inference and Applications to Clustering. John Wiley and Sons, 1988. • P. Michaud. Clustering techniques. Future Generation Computer systems, 13, 1997. • R. Ng and J. Han. Efficient and effective clustering method for spatial data mining. VLDB'94. • E. Schikuta. Grid clustering: An efficient hierarchical clustering method for very large data sets. Proc. 1996 Int. Conf. on Pattern Recognition, 101-105. • G. Sheikholeslami, S. Chatterjee, and A. Zhang. WaveCluster: A multi-resolution clustering approach for very large spatial databases. VLDB’98. • W. Wang, Yang, R. Muntz, STING: A Statistical Information grid Approach to Spatial Data Mining, VLDB’97. • T. Zhang, R. Ramakrishnan, and M. Livny. BIRCH : an efficient data clustering method for very large databases. SIGMOD'96.